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Calculus and Analytic Geometry — One Question at a Time

Calculus and Analytic Geometry — One Question at a Time

Practise 150 Calculus and Analytic Geometry multiple-choice questions covering limits, derivatives, applications of differentiation, integrals, analytic geometry and vectors. Work through one question at a time, check the correct answer with an explanation, and select Next Question to continue. This is self-paced practice without automatic scoring.

Before you practise: Read Calculus and Analytic Geometry study material and examples. You can also browse all 1,576 interactive MCQs.

Practice Set 1 · Question 1 of 150

Question 1. Evaluate lim(x→0) sin x/x for x in radians.

  1. 0
  2. 1
  3. ∞
  4. −1
Show Answer and Explanation

Correct answer: B. 1

The fundamental trigonometric limit equals 1.

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Practice Set 1 · Question 2 of 150

Question 2. Evaluate lim(x→∞) (3x²+1)/(x²+2).

  1. 0
  2. 1
  3. 2
  4. 3
Show Answer and Explanation

Correct answer: D. 3

Take the ratio of leading coefficients.

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Practice Set 1 · Question 3 of 150

Question 3. What is the derivative of ln x for x>0?

  1. 1/x
  2. ln x
  3. x
  4. eˣ
Show Answer and Explanation

Correct answer: A. 1/x

d(ln x)/dx=1/x.

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Practice Set 1 · Question 4 of 150

Question 4. Differentiate (2x+1)⁵.

  1. 5(2x+1)⁴
  2. 10(2x+1)⁴
  3. (2x+1)⁶
  4. 10x⁴
Show Answer and Explanation

Correct answer: B. 10(2x+1)⁴

Differentiate the outer fifth power and multiply by the inner derivative 2.

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Practice Set 1 · Question 5 of 150

Question 5. For x²+y²=25, find dy/dx where y≠0.

  1. x/y
  2. −y/x
  3. −x/y
  4. 2x+2y
Show Answer and Explanation

Correct answer: C. −x/y

Differentiate to get 2x+2yy’=0, then solve for y’.

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Practice Set 1 · Question 6 of 150

Question 6. If f'(x)>0 throughout an interval, what is f on that interval?

  1. Decreasing
  2. Constant
  3. Undefined
  4. Increasing
Show Answer and Explanation

Correct answer: D. Increasing

A positive derivative gives strict increase.

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Practice Set 1 · Question 7 of 150

Question 7. For f(x)=x³, find f”(x).

  1. 6x
  2. 3x²
  3. x²
  4. 6
Show Answer and Explanation

Correct answer: A. 6x

Differentiate twice: 3x², then 6x.

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Practice Set 1 · Question 8 of 150

Question 8. Evaluate ∫cos x dx.

  1. −sin x+C
  2. sin x+C
  3. tan x+C
  4. −cos x+C
Show Answer and Explanation

Correct answer: B. sin x+C

The derivative of sin x is cos x.

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Practice Set 1 · Question 9 of 150

Question 9. Does ∫₁∞ 1/x² dx converge?

  1. No
  2. Only conditionally
  3. Yes, to 1
  4. Yes, to infinity
Show Answer and Explanation

Correct answer: C. Yes, to 1

Integrate to 1−1/R then let R tend to infinity.

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Practice Set 1 · Question 10 of 150

Question 10. Which is a normal vector to the plane 2x−3y+z=5?

  1. (1,1,1)
  2. (2,3,1)
  3. (5,0,0)
  4. (2,−3,1)
Show Answer and Explanation

Correct answer: D. (2,−3,1)

Coefficients of x,y,z form a plane normal.

Continue to Practice Set 2

Practice Set 2 · Question 11 of 150

Question 11. Which function is continuous at every real number?

  1. f(x)=1/x
  2. f(x)=x²+1
  3. f(x)=1/(x−2)
  4. f(x)=tan x
Show Answer and Explanation

Correct answer: B. f(x)=x²+1

Every polynomial is continuous on the real line.

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Practice Set 2 · Question 12 of 150

Question 12. Differentiate x eˣ.

  1. eˣ
  2. xeˣ
  3. (x+1)eˣ
  4. (x−1)eˣ
Show Answer and Explanation

Correct answer: C. (x+1)eˣ

The product rule gives eˣ+xeˣ.

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Practice Set 2 · Question 13 of 150

Question 13. Differentiate 1/x for x≠0.

  1. 1/x²
  2. −1/x²
  3. −x²
  4. ln|x|
Show Answer and Explanation

Correct answer: B. −1/x²

Write x⁻¹ and apply the power rule.

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Practice Set 2 · Question 14 of 150

Question 14. For y=x², which line is tangent at x=1?

  1. y=2x−1
  2. y=x−1
  3. y=2x+1
  4. y=x²+1
Show Answer and Explanation

Correct answer: A. y=2x−1

Point is (1,1) and slope is 2, so y−1=2(x−1).

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Practice Set 2 · Question 15 of 150

Question 15. What is the minimum value of x²−4x+7 over real x?

  1. −3
  2. 2
  3. 3
  4. 7
Show Answer and Explanation

Correct answer: C. 3

Complete the square: (x−2)²+3, minimized at x=2.

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Practice Set 2 · Question 16 of 150

Question 16. Evaluate ∫2x cos(x²)dx.

  1. cos(x²)+C
  2. 2sin x+C
  3. x² sin(x²)+C
  4. sin(x²)+C
Show Answer and Explanation

Correct answer: D. sin(x²)+C

Substitute u=x², du=2x dx.

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Practice Set 2 · Question 17 of 150

Question 17. What is the area under y=x from x=0 to x=2?

  1. 1
  2. 2
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: B. 2

Integral of x from 0 to 2 is [x²/2]₀²=2.

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Practice Set 2 · Question 18 of 150

Question 18. For p>1, what is ∫₁∞ x⁻ᵖ dx?

  1. 1/(p−1)
  2. 1/(1−p)
  3. p−1
  4. ∞
Show Answer and Explanation

Correct answer: A. 1/(p−1)

Integrate and take the limit at infinity; the result is 1/(p−1).

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Practice Set 2 · Question 19 of 150

Question 19. Which vector is parallel to the line (x,y,z)=(1,2,3)+t(2,−1,4)?

  1. (1,2,3)
  2. (0,0,1)
  3. (2,−1,4)
  4. (2,1,4)
Show Answer and Explanation

Correct answer: C. (2,−1,4)

The coefficient vector of t is the direction vector.

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Practice Set 2 · Question 20 of 150

Question 20. For f(x)=x² on [1,3], which c satisfies f'(c)=[f(3)−f(1)]/(3−1)?

  1. 1
  2. 2
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: B. 2

Average slope is (9−1)/2=4, so 2c=4 and c=2.

Continue to Practice Set 3

Practice Set 3 · Question 21 of 150

Question 21. Find lim(x→0) (1−cos x)/x² in radians.

  1. 0
  2. 1/2
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: B. 1/2

The standard trigonometric limit is 1/2.

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Practice Set 3 · Question 22 of 150

Question 22. Differentiate arctan x.

  1. 1/(1+x²)
  2. 1/(1−x²)
  3. −1/(1+x²)
  4. arctan x
Show Answer and Explanation

Correct answer: A. 1/(1+x²)

Derivative of arctan x equals 1/(1+x²).

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Practice Set 3 · Question 23 of 150

Question 23. Differentiate ln(x²+1).

  1. 1/(x²+1)
  2. 2x
  3. 2x/(x²+1)
  4. x²+1
Show Answer and Explanation

Correct answer: C. 2x/(x²+1)

Apply the logarithm derivative and the chain rule.

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Practice Set 3 · Question 24 of 150

Question 24. What does the Mean Value Theorem guarantee under its hypotheses?

  1. f'(c)=0 always
  2. f(a)=f(b)
  3. f is linear
  4. f'(c) equals the average slope for some c
Show Answer and Explanation

Correct answer: D. f'(c) equals the average slope for some c

There exists c in (a,b) with f'(c)=[f(b)−f(a)]/(b−a).

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Practice Set 3 · Question 25 of 150

Question 25. Where does the derivative of x³−3x vanish?

  1. At x=0
  2. At x=−1 and 1
  3. At x=−3 and 3
  4. At x=1 only
Show Answer and Explanation

Correct answer: B. At x=−1 and 1

3x²−3=0 gives x²=1.

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Practice Set 3 · Question 26 of 150

Question 26. Find ∫sec²x dx.

  1. tan x+C
  2. sec x+C
  3. sin x+C
  4. −cot x+C
Show Answer and Explanation

Correct answer: A. tan x+C

Derivative of tangent is secant squared.

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Practice Set 3 · Question 27 of 150

Question 27. Find the area between y=2 and y=x on [0,2].

  1. 1
  2. 4
  3. 2
  4. 8
Show Answer and Explanation

Correct answer: C. 2

∫₀²(2−x)dx=2.

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Practice Set 3 · Question 28 of 150

Question 28. For what values of p does ∫₁∞ x^(−p)dx converge?

  1. p<0
  2. p=0
  3. p≤1
  4. p>1
Show Answer and Explanation

Correct answer: D. p>1

The integral converges exactly when p>1.

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Practice Set 3 · Question 29 of 150

Question 29. Find the plane through (0,0,3) with normal (1,2,−1).

  1. x+2y−z=3
  2. x+2y−z=−3
  3. x−2y+z=3
  4. x+y+z=3
Show Answer and Explanation

Correct answer: B. x+2y−z=−3

The normal equation gives x+2y−(z−3)=0.

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Practice Set 3 · Question 30 of 150

Question 30. What is the slope of y=sin x at x=0?

  1. 1
  2. 0
  3. −1
  4. π
Show Answer and Explanation

Correct answer: A. 1

The derivative is cos x, giving cos 0=1.

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Practice Set 4 · Question 31 of 150

Question 31. Evaluate lim(x→1) (2x+3).

  1. 6
  2. 4
  3. 8
  4. 5
Show Answer and Explanation

Correct answer: D. 5

Polynomials are continuous, so substitute x=1 to obtain 5.

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Practice Set 4 · Question 32 of 150

Question 32. Evaluate lim(x→2) (3x+4).

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

Polynomials are continuous, so substitute x=2 to obtain 10.

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Practice Set 4 · Question 33 of 150

Question 33. Evaluate lim(x→3) (4x+5).

  1. 20
  2. 17
  3. 18
  4. 16
Show Answer and Explanation

Correct answer: B. 17

Polynomials are continuous, so substitute x=3 to obtain 17.

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Practice Set 4 · Question 34 of 150

Question 34. Evaluate lim(x→4) (5x+6).

  1. 25
  2. 29
  3. 26
  4. 27
Show Answer and Explanation

Correct answer: C. 26

Polynomials are continuous, so substitute x=4 to obtain 26.

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Practice Set 4 · Question 35 of 150

Question 35. Evaluate lim(x→5) (6x+7).

  1. 38
  2. 36
  3. 40
  4. 37
Show Answer and Explanation

Correct answer: D. 37

Polynomials are continuous, so substitute x=5 to obtain 37.

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Practice Set 4 · Question 36 of 150

Question 36. Evaluate lim(x→6) (7x+8).

  1. 50
  2. 51
  3. 49
  4. 53
Show Answer and Explanation

Correct answer: A. 50

Polynomials are continuous, so substitute x=6 to obtain 50.

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Practice Set 4 · Question 37 of 150

Question 37. Evaluate lim(x→7) (8x+9).

  1. 68
  2. 65
  3. 66
  4. 64
Show Answer and Explanation

Correct answer: B. 65

Polynomials are continuous, so substitute x=7 to obtain 65.

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Practice Set 4 · Question 38 of 150

Question 38. Evaluate lim(x→8) (9x+10).

  1. 81
  2. 85
  3. 82
  4. 83
Show Answer and Explanation

Correct answer: C. 82

Polynomials are continuous, so substitute x=8 to obtain 82.

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Practice Set 4 · Question 39 of 150

Question 39. Evaluate lim(x→9) (10x+11).

  1. 102
  2. 100
  3. 104
  4. 101
Show Answer and Explanation

Correct answer: D. 101

Polynomials are continuous, so substitute x=9 to obtain 101.

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Practice Set 4 · Question 40 of 150

Question 40. Evaluate lim(x→10) (11x+12).

  1. 122
  2. 123
  3. 121
  4. 125
Show Answer and Explanation

Correct answer: A. 122

Polynomials are continuous, so substitute x=10 to obtain 122.

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Practice Set 5 · Question 41 of 150

Question 41. Find the derivative at x=1 of f(x)=2x².

  1. 7
  2. 4
  3. 5
  4. 3
Show Answer and Explanation

Correct answer: B. 4

f'(x)=4x, so f'(1)=4.

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Practice Set 5 · Question 42 of 150

Question 42. Find the derivative at x=2 of f(x)=3x².

  1. 11
  2. 15
  3. 12
  4. 13
Show Answer and Explanation

Correct answer: C. 12

f'(x)=6x, so f'(2)=12.

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Practice Set 5 · Question 43 of 150

Question 43. Find the derivative at x=3 of f(x)=4x².

  1. 25
  2. 23
  3. 27
  4. 24
Show Answer and Explanation

Correct answer: D. 24

f'(x)=8x, so f'(3)=24.

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Practice Set 5 · Question 44 of 150

Question 44. Find the derivative at x=4 of f(x)=5x².

  1. 40
  2. 41
  3. 39
  4. 43
Show Answer and Explanation

Correct answer: A. 40

f'(x)=10x, so f'(4)=40.

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Practice Set 5 · Question 45 of 150

Question 45. Find the derivative at x=5 of f(x)=6x².

  1. 63
  2. 60
  3. 61
  4. 59
Show Answer and Explanation

Correct answer: B. 60

f'(x)=12x, so f'(5)=60.

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Practice Set 5 · Question 46 of 150

Question 46. Find the derivative at x=6 of f(x)=7x².

  1. 83
  2. 87
  3. 84
  4. 85
Show Answer and Explanation

Correct answer: C. 84

f'(x)=14x, so f'(6)=84.

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Practice Set 5 · Question 47 of 150

Question 47. Find the derivative at x=7 of f(x)=8x².

  1. 113
  2. 111
  3. 115
  4. 112
Show Answer and Explanation

Correct answer: D. 112

f'(x)=16x, so f'(7)=112.

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Practice Set 5 · Question 48 of 150

Question 48. Find the derivative at x=8 of f(x)=9x².

  1. 144
  2. 145
  3. 143
  4. 147
Show Answer and Explanation

Correct answer: A. 144

f'(x)=18x, so f'(8)=144.

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Practice Set 5 · Question 49 of 150

Question 49. Find the derivative at x=9 of f(x)=10x².

  1. 183
  2. 180
  3. 181
  4. 179
Show Answer and Explanation

Correct answer: B. 180

f'(x)=20x, so f'(9)=180.

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Practice Set 5 · Question 50 of 150

Question 50. Find the derivative at x=10 of f(x)=11x².

  1. 219
  2. 223
  3. 220
  4. 221
Show Answer and Explanation

Correct answer: C. 220

f'(x)=22x, so f'(10)=220.

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Practice Set 6 · Question 51 of 150

Question 51. Compute ∫₀¹ 2x^1 dx.

  1. 2
  2. 0
  3. 4
  4. 1
Show Answer and Explanation

Correct answer: D. 1

Using the power rule, the definite integral equals 2/(2)=1.

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Practice Set 6 · Question 52 of 150

Question 52. Compute ∫₀¹ 3x^2 dx.

  1. 1
  2. 2
  3. 0
  4. 4
Show Answer and Explanation

Correct answer: A. 1

Using the power rule, the definite integral equals 3/(3)=1.

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Practice Set 6 · Question 53 of 150

Question 53. Compute ∫₀¹ 4x^3 dx.

  1. 4
  2. 1
  3. 2
  4. 0
Show Answer and Explanation

Correct answer: B. 1

Using the power rule, the definite integral equals 4/(4)=1.

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Practice Set 6 · Question 54 of 150

Question 54. Compute ∫₀¹ 5x^4 dx.

  1. 0
  2. 4
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: C. 1

Using the power rule, the definite integral equals 5/(5)=1.

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Practice Set 6 · Question 55 of 150

Question 55. Compute ∫₀¹ 6x^5 dx.

  1. 2
  2. 0
  3. 4
  4. 1
Show Answer and Explanation

Correct answer: D. 1

Using the power rule, the definite integral equals 6/(6)=1.

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Practice Set 6 · Question 56 of 150

Question 56. Compute ∫₀¹ 7x^6 dx.

  1. 1
  2. 2
  3. 0
  4. 4
Show Answer and Explanation

Correct answer: A. 1

Using the power rule, the definite integral equals 7/(7)=1.

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Practice Set 6 · Question 57 of 150

Question 57. Compute ∫₀¹ 8x^7 dx.

  1. 4
  2. 1
  3. 2
  4. 0
Show Answer and Explanation

Correct answer: B. 1

Using the power rule, the definite integral equals 8/(8)=1.

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Practice Set 6 · Question 58 of 150

Question 58. Compute ∫₀¹ 9x^8 dx.

  1. 0
  2. 4
  3. 1
  4. 2
Show Answer and Explanation

Correct answer: C. 1

Using the power rule, the definite integral equals 9/(9)=1.

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Practice Set 6 · Question 59 of 150

Question 59. Compute ∫₀¹ 10x^9 dx.

  1. 2
  2. 0
  3. 4
  4. 1
Show Answer and Explanation

Correct answer: D. 1

Using the power rule, the definite integral equals 10/(10)=1.

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Practice Set 6 · Question 60 of 150

Question 60. Compute ∫₀¹ 11x^10 dx.

  1. 1
  2. 2
  3. 0
  4. 4
Show Answer and Explanation

Correct answer: A. 1

Using the power rule, the definite integral equals 11/(11)=1.

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Practice Set 7 · Question 61 of 150

Question 61. For f(x)=2x+3, what is the slope of its graph?

  1. 5
  2. 2
  3. 3
  4. 1
Show Answer and Explanation

Correct answer: B. 2

In y=mx+b, the coefficient of x is the slope m=2.

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Practice Set 7 · Question 62 of 150

Question 62. For f(x)=3x+4, what is the slope of its graph?

  1. 2
  2. 6
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

In y=mx+b, the coefficient of x is the slope m=3.

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Practice Set 7 · Question 63 of 150

Question 63. For f(x)=4x+5, what is the slope of its graph?

  1. 5
  2. 3
  3. 7
  4. 4
Show Answer and Explanation

Correct answer: D. 4

In y=mx+b, the coefficient of x is the slope m=4.

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Practice Set 7 · Question 64 of 150

Question 64. For f(x)=5x+6, what is the slope of its graph?

  1. 5
  2. 6
  3. 4
  4. 8
Show Answer and Explanation

Correct answer: A. 5

In y=mx+b, the coefficient of x is the slope m=5.

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Practice Set 7 · Question 65 of 150

Question 65. For f(x)=6x+7, what is the slope of its graph?

  1. 9
  2. 6
  3. 7
  4. 5
Show Answer and Explanation

Correct answer: B. 6

In y=mx+b, the coefficient of x is the slope m=6.

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Practice Set 7 · Question 66 of 150

Question 66. For f(x)=7x+8, what is the slope of its graph?

  1. 6
  2. 10
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

In y=mx+b, the coefficient of x is the slope m=7.

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Practice Set 7 · Question 67 of 150

Question 67. For f(x)=8x+9, what is the slope of its graph?

  1. 9
  2. 7
  3. 11
  4. 8
Show Answer and Explanation

Correct answer: D. 8

In y=mx+b, the coefficient of x is the slope m=8.

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Practice Set 7 · Question 68 of 150

Question 68. For f(x)=9x+10, what is the slope of its graph?

  1. 9
  2. 10
  3. 8
  4. 12
Show Answer and Explanation

Correct answer: A. 9

In y=mx+b, the coefficient of x is the slope m=9.

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Practice Set 7 · Question 69 of 150

Question 69. For f(x)=10x+11, what is the slope of its graph?

  1. 13
  2. 10
  3. 11
  4. 9
Show Answer and Explanation

Correct answer: B. 10

In y=mx+b, the coefficient of x is the slope m=10.

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Practice Set 7 · Question 70 of 150

Question 70. For f(x)=11x+12, what is the slope of its graph?

  1. 10
  2. 14
  3. 11
  4. 12
Show Answer and Explanation

Correct answer: C. 11

In y=mx+b, the coefficient of x is the slope m=11.

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Practice Set 8 · Question 71 of 150

Question 71. Find the radius of the circle (x−1)²+(y+2)²=4.

  1. 3
  2. 1
  3. 5
  4. 2
Show Answer and Explanation

Correct answer: D. 2

The right-hand side is r²=4; hence radius r=2.

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Practice Set 8 · Question 72 of 150

Question 72. Find the radius of the circle (x−2)²+(y+3)²=9.

  1. 3
  2. 4
  3. 2
  4. 6
Show Answer and Explanation

Correct answer: A. 3

The right-hand side is r²=9; hence radius r=3.

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Practice Set 8 · Question 73 of 150

Question 73. Find the radius of the circle (x−3)²+(y+4)²=16.

  1. 7
  2. 4
  3. 5
  4. 3
Show Answer and Explanation

Correct answer: B. 4

The right-hand side is r²=16; hence radius r=4.

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Practice Set 8 · Question 74 of 150

Question 74. Find the radius of the circle (x−4)²+(y+5)²=25.

  1. 4
  2. 8
  3. 5
  4. 6
Show Answer and Explanation

Correct answer: C. 5

The right-hand side is r²=25; hence radius r=5.

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Practice Set 8 · Question 75 of 150

Question 75. Find the radius of the circle (x−5)²+(y+6)²=36.

  1. 7
  2. 5
  3. 9
  4. 6
Show Answer and Explanation

Correct answer: D. 6

The right-hand side is r²=36; hence radius r=6.

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Practice Set 8 · Question 76 of 150

Question 76. Find the radius of the circle (x−6)²+(y+7)²=49.

  1. 7
  2. 8
  3. 6
  4. 10
Show Answer and Explanation

Correct answer: A. 7

The right-hand side is r²=49; hence radius r=7.

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Practice Set 8 · Question 77 of 150

Question 77. Find the radius of the circle (x−7)²+(y+8)²=64.

  1. 11
  2. 8
  3. 9
  4. 7
Show Answer and Explanation

Correct answer: B. 8

The right-hand side is r²=64; hence radius r=8.

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Practice Set 8 · Question 78 of 150

Question 78. Find the radius of the circle (x−8)²+(y+9)²=81.

  1. 8
  2. 12
  3. 9
  4. 10
Show Answer and Explanation

Correct answer: C. 9

The right-hand side is r²=81; hence radius r=9.

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Practice Set 8 · Question 79 of 150

Question 79. Find the radius of the circle (x−9)²+(y+10)²=100.

  1. 11
  2. 9
  3. 13
  4. 10
Show Answer and Explanation

Correct answer: D. 10

The right-hand side is r²=100; hence radius r=10.

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Practice Set 8 · Question 80 of 150

Question 80. Find the radius of the circle (x−10)²+(y+11)²=121.

  1. 11
  2. 12
  3. 10
  4. 14
Show Answer and Explanation

Correct answer: A. 11

The right-hand side is r²=121; hence radius r=11.

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Practice Set 9 · Question 81 of 150

Question 81. Evaluate the second derivative of f(x)=2x³ at x=1.

  1. 15
  2. 12
  3. 13
  4. 11
Show Answer and Explanation

Correct answer: B. 12

Differentiate twice: f”(x)=12x; at 1 this is 12.

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Practice Set 9 · Question 82 of 150

Question 82. Evaluate the second derivative of f(x)=3x³ at x=2.

  1. 35
  2. 39
  3. 36
  4. 37
Show Answer and Explanation

Correct answer: C. 36

Differentiate twice: f”(x)=18x; at 2 this is 36.

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Practice Set 9 · Question 83 of 150

Question 83. Evaluate the second derivative of f(x)=4x³ at x=3.

  1. 73
  2. 71
  3. 75
  4. 72
Show Answer and Explanation

Correct answer: D. 72

Differentiate twice: f”(x)=24x; at 3 this is 72.

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Practice Set 9 · Question 84 of 150

Question 84. Evaluate the second derivative of f(x)=5x³ at x=4.

  1. 120
  2. 121
  3. 119
  4. 123
Show Answer and Explanation

Correct answer: A. 120

Differentiate twice: f”(x)=30x; at 4 this is 120.

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Practice Set 9 · Question 85 of 150

Question 85. Evaluate the second derivative of f(x)=6x³ at x=5.

  1. 183
  2. 180
  3. 181
  4. 179
Show Answer and Explanation

Correct answer: B. 180

Differentiate twice: f”(x)=36x; at 5 this is 180.

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Practice Set 9 · Question 86 of 150

Question 86. Evaluate the second derivative of f(x)=7x³ at x=6.

  1. 251
  2. 255
  3. 252
  4. 253
Show Answer and Explanation

Correct answer: C. 252

Differentiate twice: f”(x)=42x; at 6 this is 252.

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Practice Set 9 · Question 87 of 150

Question 87. Evaluate the second derivative of f(x)=8x³ at x=7.

  1. 337
  2. 335
  3. 339
  4. 336
Show Answer and Explanation

Correct answer: D. 336

Differentiate twice: f”(x)=48x; at 7 this is 336.

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Practice Set 9 · Question 88 of 150

Question 88. Evaluate the second derivative of f(x)=9x³ at x=8.

  1. 432
  2. 433
  3. 431
  4. 435
Show Answer and Explanation

Correct answer: A. 432

Differentiate twice: f”(x)=54x; at 8 this is 432.

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Practice Set 9 · Question 89 of 150

Question 89. Evaluate the second derivative of f(x)=10x³ at x=9.

  1. 543
  2. 540
  3. 541
  4. 539
Show Answer and Explanation

Correct answer: B. 540

Differentiate twice: f”(x)=60x; at 9 this is 540.

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Practice Set 9 · Question 90 of 150

Question 90. Evaluate the second derivative of f(x)=11x³ at x=10.

  1. 659
  2. 663
  3. 660
  4. 661
Show Answer and Explanation

Correct answer: C. 660

Differentiate twice: f”(x)=66x; at 10 this is 660.

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Practice Set 10 · Question 91 of 150

Question 91. Find ∫₀^2 2x dx.

  1. 5
  2. 3
  3. 7
  4. 4
Show Answer and Explanation

Correct answer: D. 4

The antiderivative is x²; evaluating at 2 and 0 gives 4.

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Practice Set 10 · Question 92 of 150

Question 92. Find ∫₀^3 2x dx.

  1. 9
  2. 10
  3. 8
  4. 12
Show Answer and Explanation

Correct answer: A. 9

The antiderivative is x²; evaluating at 3 and 0 gives 9.

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Practice Set 10 · Question 93 of 150

Question 93. Find ∫₀^4 2x dx.

  1. 19
  2. 16
  3. 17
  4. 15
Show Answer and Explanation

Correct answer: B. 16

The antiderivative is x²; evaluating at 4 and 0 gives 16.

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Practice Set 10 · Question 94 of 150

Question 94. Find ∫₀^5 2x dx.

  1. 24
  2. 28
  3. 25
  4. 26
Show Answer and Explanation

Correct answer: C. 25

The antiderivative is x²; evaluating at 5 and 0 gives 25.

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Practice Set 10 · Question 95 of 150

Question 95. Find ∫₀^6 2x dx.

  1. 37
  2. 35
  3. 39
  4. 36
Show Answer and Explanation

Correct answer: D. 36

The antiderivative is x²; evaluating at 6 and 0 gives 36.

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Practice Set 10 · Question 96 of 150

Question 96. Find ∫₀^7 2x dx.

  1. 49
  2. 50
  3. 48
  4. 52
Show Answer and Explanation

Correct answer: A. 49

The antiderivative is x²; evaluating at 7 and 0 gives 49.

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Practice Set 10 · Question 97 of 150

Question 97. Find ∫₀^8 2x dx.

  1. 67
  2. 64
  3. 65
  4. 63
Show Answer and Explanation

Correct answer: B. 64

The antiderivative is x²; evaluating at 8 and 0 gives 64.

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Practice Set 10 · Question 98 of 150

Question 98. Find ∫₀^9 2x dx.

  1. 80
  2. 84
  3. 81
  4. 82
Show Answer and Explanation

Correct answer: C. 81

The antiderivative is x²; evaluating at 9 and 0 gives 81.

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Practice Set 10 · Question 99 of 150

Question 99. Find ∫₀^10 2x dx.

  1. 101
  2. 99
  3. 103
  4. 100
Show Answer and Explanation

Correct answer: D. 100

The antiderivative is x²; evaluating at 10 and 0 gives 100.

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Practice Set 10 · Question 100 of 150

Question 100. Find ∫₀^11 2x dx.

  1. 121
  2. 122
  3. 120
  4. 124
Show Answer and Explanation

Correct answer: A. 121

The antiderivative is x²; evaluating at 11 and 0 gives 121.

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Practice Set 11 · Question 101 of 150

Question 101. Solve 2x=6 for x.

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

Divide both sides by 2 to obtain x=3.

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Practice Set 11 · Question 102 of 150

Question 102. Solve 3x=12 for x.

  1. 3
  2. 7
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

Divide both sides by 3 to obtain x=4.

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Practice Set 11 · Question 103 of 150

Question 103. Solve 4x=20 for x.

  1. 6
  2. 4
  3. 8
  4. 5
Show Answer and Explanation

Correct answer: D. 5

Divide both sides by 4 to obtain x=5.

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Practice Set 11 · Question 104 of 150

Question 104. Solve 5x=30 for x.

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

Divide both sides by 5 to obtain x=6.

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Practice Set 11 · Question 105 of 150

Question 105. Solve 6x=42 for x.

  1. 10
  2. 7
  3. 8
  4. 6
Show Answer and Explanation

Correct answer: B. 7

Divide both sides by 6 to obtain x=7.

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Practice Set 11 · Question 106 of 150

Question 106. Solve 7x=56 for x.

  1. 7
  2. 11
  3. 8
  4. 9
Show Answer and Explanation

Correct answer: C. 8

Divide both sides by 7 to obtain x=8.

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Practice Set 11 · Question 107 of 150

Question 107. Solve 8x=72 for x.

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

Divide both sides by 8 to obtain x=9.

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Practice Set 11 · Question 108 of 150

Question 108. Solve 9x=90 for x.

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

Divide both sides by 9 to obtain x=10.

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Practice Set 11 · Question 109 of 150

Question 109. Solve 10x=110 for x.

  1. 14
  2. 11
  3. 12
  4. 10
Show Answer and Explanation

Correct answer: B. 11

Divide both sides by 10 to obtain x=11.

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Practice Set 11 · Question 110 of 150

Question 110. Solve 11x=132 for x.

  1. 11
  2. 15
  3. 12
  4. 13
Show Answer and Explanation

Correct answer: C. 12

Divide both sides by 11 to obtain x=12.

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Practice Set 12 · Question 111 of 150

Question 111. Find the x-intercept of y=2x−6.

  1. 4
  2. 2
  3. 6
  4. 3
Show Answer and Explanation

Correct answer: D. 3

Set y=0 and solve 2x=6, giving x=3.

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Practice Set 12 · Question 112 of 150

Question 112. Find the x-intercept of y=3x−12.

  1. 4
  2. 5
  3. 3
  4. 7
Show Answer and Explanation

Correct answer: A. 4

Set y=0 and solve 3x=12, giving x=4.

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Practice Set 12 · Question 113 of 150

Question 113. Find the x-intercept of y=4x−20.

  1. 8
  2. 5
  3. 6
  4. 4
Show Answer and Explanation

Correct answer: B. 5

Set y=0 and solve 4x=20, giving x=5.

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Practice Set 12 · Question 114 of 150

Question 114. Find the x-intercept of y=5x−30.

  1. 5
  2. 9
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: C. 6

Set y=0 and solve 5x=30, giving x=6.

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Practice Set 12 · Question 115 of 150

Question 115. Find the x-intercept of y=6x−42.

  1. 8
  2. 6
  3. 10
  4. 7
Show Answer and Explanation

Correct answer: D. 7

Set y=0 and solve 6x=42, giving x=7.

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Practice Set 12 · Question 116 of 150

Question 116. Find the x-intercept of y=7x−56.

  1. 8
  2. 9
  3. 7
  4. 11
Show Answer and Explanation

Correct answer: A. 8

Set y=0 and solve 7x=56, giving x=8.

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Practice Set 12 · Question 117 of 150

Question 117. Find the x-intercept of y=8x−72.

  1. 12
  2. 9
  3. 10
  4. 8
Show Answer and Explanation

Correct answer: B. 9

Set y=0 and solve 8x=72, giving x=9.

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Practice Set 12 · Question 118 of 150

Question 118. Find the x-intercept of y=9x−90.

  1. 9
  2. 13
  3. 10
  4. 11
Show Answer and Explanation

Correct answer: C. 10

Set y=0 and solve 9x=90, giving x=10.

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Practice Set 12 · Question 119 of 150

Question 119. Find the x-intercept of y=10x−110.

  1. 12
  2. 10
  3. 14
  4. 11
Show Answer and Explanation

Correct answer: D. 11

Set y=0 and solve 10x=110, giving x=11.

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Practice Set 12 · Question 120 of 150

Question 120. Find the x-intercept of y=11x−132.

  1. 12
  2. 13
  3. 11
  4. 15
Show Answer and Explanation

Correct answer: A. 12

Set y=0 and solve 11x=132, giving x=12.

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Practice Set 13 · Question 121 of 150

Question 121. Find the area of a triangle with base 2 and height 3.

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

Area=(1/2)×base×height=3.

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Practice Set 13 · Question 122 of 150

Question 122. Find the area of a triangle with base 3 and height 4.

  1. 5
  2. 9
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: C. 6

Area=(1/2)×base×height=6.

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Practice Set 13 · Question 123 of 150

Question 123. Find the area of a triangle with base 4 and height 5.

  1. 11
  2. 9
  3. 13
  4. 10
Show Answer and Explanation

Correct answer: D. 10

Area=(1/2)×base×height=10.

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Practice Set 13 · Question 124 of 150

Question 124. Find the area of a triangle with base 5 and height 6.

  1. 15
  2. 16
  3. 14
  4. 18
Show Answer and Explanation

Correct answer: A. 15

Area=(1/2)×base×height=15.

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Practice Set 13 · Question 125 of 150

Question 125. Find the area of a triangle with base 6 and height 7.

  1. 24
  2. 21
  3. 22
  4. 20
Show Answer and Explanation

Correct answer: B. 21

Area=(1/2)×base×height=21.

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Practice Set 13 · Question 126 of 150

Question 126. Find the area of a triangle with base 7 and height 8.

  1. 27
  2. 31
  3. 28
  4. 29
Show Answer and Explanation

Correct answer: C. 28

Area=(1/2)×base×height=28.

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Practice Set 13 · Question 127 of 150

Question 127. Find the area of a triangle with base 8 and height 9.

  1. 37
  2. 35
  3. 39
  4. 36
Show Answer and Explanation

Correct answer: D. 36

Area=(1/2)×base×height=36.

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Practice Set 13 · Question 128 of 150

Question 128. Find the area of a triangle with base 9 and height 10.

  1. 45
  2. 46
  3. 44
  4. 48
Show Answer and Explanation

Correct answer: A. 45

Area=(1/2)×base×height=45.

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Practice Set 13 · Question 129 of 150

Question 129. Find the area of a triangle with base 10 and height 11.

  1. 58
  2. 55
  3. 56
  4. 54
Show Answer and Explanation

Correct answer: B. 55

Area=(1/2)×base×height=55.

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Practice Set 13 · Question 130 of 150

Question 130. Find the area of a triangle with base 11 and height 12.

  1. 65
  2. 69
  3. 66
  4. 67
Show Answer and Explanation

Correct answer: C. 66

Area=(1/2)×base×height=66.

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Practice Set 14 · Question 131 of 150

Question 131. Evaluate the derivative of f(x)=x^2 at x=1.

  1. 3
  2. 1
  3. 5
  4. 2
Show Answer and Explanation

Correct answer: D. 2

By the power rule, f'(x)=2x^1, hence f'(1)=2.

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Practice Set 14 · Question 132 of 150

Question 132. Evaluate the derivative of f(x)=x^3 at x=1.

  1. 3
  2. 4
  3. 2
  4. 6
Show Answer and Explanation

Correct answer: A. 3

By the power rule, f'(x)=3x^2, hence f'(1)=3.

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Practice Set 14 · Question 133 of 150

Question 133. Evaluate the derivative of f(x)=x^4 at x=1.

  1. 7
  2. 4
  3. 5
  4. 3
Show Answer and Explanation

Correct answer: B. 4

By the power rule, f'(x)=4x^3, hence f'(1)=4.

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Practice Set 14 · Question 134 of 150

Question 134. Evaluate the derivative of f(x)=x^5 at x=1.

  1. 4
  2. 8
  3. 5
  4. 6
Show Answer and Explanation

Correct answer: C. 5

By the power rule, f'(x)=5x^4, hence f'(1)=5.

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Practice Set 14 · Question 135 of 150

Question 135. Evaluate the derivative of f(x)=x^6 at x=1.

  1. 7
  2. 5
  3. 9
  4. 6
Show Answer and Explanation

Correct answer: D. 6

By the power rule, f'(x)=6x^5, hence f'(1)=6.

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Practice Set 14 · Question 136 of 150

Question 136. Evaluate the derivative of f(x)=x^7 at x=1.

  1. 7
  2. 8
  3. 6
  4. 10
Show Answer and Explanation

Correct answer: A. 7

By the power rule, f'(x)=7x^6, hence f'(1)=7.

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Practice Set 14 · Question 137 of 150

Question 137. Evaluate the derivative of f(x)=x^8 at x=1.

  1. 11
  2. 8
  3. 9
  4. 7
Show Answer and Explanation

Correct answer: B. 8

By the power rule, f'(x)=8x^7, hence f'(1)=8.

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Practice Set 14 · Question 138 of 150

Question 138. Evaluate the derivative of f(x)=x^9 at x=1.

  1. 8
  2. 12
  3. 9
  4. 10
Show Answer and Explanation

Correct answer: C. 9

By the power rule, f'(x)=9x^8, hence f'(1)=9.

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Practice Set 14 · Question 139 of 150

Question 139. Evaluate the derivative of f(x)=x^10 at x=1.

  1. 11
  2. 9
  3. 13
  4. 10
Show Answer and Explanation

Correct answer: D. 10

By the power rule, f'(x)=10x^9, hence f'(1)=10.

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Practice Set 14 · Question 140 of 150

Question 140. Evaluate the derivative of f(x)=x^11 at x=1.

  1. 11
  2. 12
  3. 10
  4. 14
Show Answer and Explanation

Correct answer: A. 11

By the power rule, f'(x)=11x^10, hence f'(1)=11.

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Practice Set 15 · Question 141 of 150

Question 141. For f(x)=(x−2)²+3, find the minimum value on R.

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

A square is nonnegative and vanishes at x=2; the minimum is 3.

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Practice Set 15 · Question 142 of 150

Question 142. For f(x)=(x−3)²+4, find the minimum value on R.

  1. 3
  2. 7
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

A square is nonnegative and vanishes at x=3; the minimum is 4.

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Practice Set 15 · Question 143 of 150

Question 143. For f(x)=(x−4)²+5, find the minimum value on R.

  1. 6
  2. 4
  3. 8
  4. 5
Show Answer and Explanation

Correct answer: D. 5

A square is nonnegative and vanishes at x=4; the minimum is 5.

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Practice Set 15 · Question 144 of 150

Question 144. For f(x)=(x−5)²+6, find the minimum value on R.

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

A square is nonnegative and vanishes at x=5; the minimum is 6.

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Practice Set 15 · Question 145 of 150

Question 145. For f(x)=(x−6)²+7, find the minimum value on R.

  1. 10
  2. 7
  3. 8
  4. 6
Show Answer and Explanation

Correct answer: B. 7

A square is nonnegative and vanishes at x=6; the minimum is 7.

Next Question →

Practice Set 15 · Question 146 of 150

Question 146. For f(x)=(x−7)²+8, find the minimum value on R.

  1. 7
  2. 11
  3. 8
  4. 9
Show Answer and Explanation

Correct answer: C. 8

A square is nonnegative and vanishes at x=7; the minimum is 8.

Next Question →

Practice Set 15 · Question 147 of 150

Question 147. For f(x)=(x−8)²+9, find the minimum value on R.

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

A square is nonnegative and vanishes at x=8; the minimum is 9.

Next Question →

Practice Set 15 · Question 148 of 150

Question 148. For f(x)=(x−9)²+10, find the minimum value on R.

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

A square is nonnegative and vanishes at x=9; the minimum is 10.

Next Question →

Practice Set 15 · Question 149 of 150

Question 149. For f(x)=(x−10)²+11, find the minimum value on R.

  1. 14
  2. 11
  3. 12
  4. 10
Show Answer and Explanation

Correct answer: B. 11

A square is nonnegative and vanishes at x=10; the minimum is 11.

Next Question →

Practice Set 15 · Question 150 of 150

Question 150. For f(x)=(x−11)²+12, find the minimum value on R.

  1. 11
  2. 15
  3. 12
  4. 13
Show Answer and Explanation

Correct answer: C. 12

A square is nonnegative and vanishes at x=11; the minimum is 12.

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