Calculus and Analytic Geometry
Introduction
Calculus and Analytic Geometry explores limits, derivatives and integrals and connects these ideas to the geometry of lines, curves and planes.
Recommended Background
Background in algebra, functions and trigonometry; Pre-Calculus I and II are useful preparation.
Topic-Wise Course Outline
Functions, Limits and Continuity
Functions and their representations; limit laws and techniques; indeterminate forms; continuous and discontinuous functions.
Differentiation
Derivatives as slopes and rates of change; product, quotient and chain rules; implicit differentiation; tangent and normal lines; linear approximation.
Applications of Derivatives
Extreme value theorem; mean value theorem; increasing and decreasing functions; maxima, minima and concavity.
Integration
Antiderivatives; substitution and other integration techniques; Riemann sums and definite integrals.
Applications and Improper Integrals
Area under and between curves; improper integrals and their convergence.
Analytic Geometry in Three Dimensions
Lines in three-dimensional space and equations of planes.
How to Study This Subject
Start with the definitions and core concepts in each topic. Work through examples, check every step of a solution, and then practise questions to test understanding.
Begin with the MCQs and worked examples below, then review each topic in the outline. More practice material will be added progressively.
Foundational MCQs and Step-by-Step Practice
Ten introductory topic-wise questions with explanations, followed by two fully worked examples. This is a starter set; further coverage is planned.
Limits
Q1. Evaluate lim(x→0) sin x/x for x in radians.
- A. 0
- B. 1
- C. ∞
- D. −1
Show correct answer and step-by-step explanation
Correct answer: B. 1
The fundamental trigonometric limit equals 1.
Q2. Evaluate lim(x→∞) (3x²+1)/(x²+2).
- A. 0
- B. 1
- C. 2
- D. 3
Show correct answer and step-by-step explanation
Correct answer: D. 3
Take the ratio of leading coefficients.
Differentiation
Q3. What is the derivative of ln x for x>0?
- A. 1/x
- B. ln x
- C. x
- D. eˣ
Show correct answer and step-by-step explanation
Correct answer: A. 1/x
d(ln x)/dx=1/x.
Chain Rule
Q4. Differentiate (2x+1)⁵.
- A. 5(2x+1)⁴
- B. 10(2x+1)⁴
- C. (2x+1)⁶
- D. 10x⁴
Show correct answer and step-by-step explanation
Correct answer: B. 10(2x+1)⁴
Differentiate the outer fifth power and multiply by the inner derivative 2.
Implicit Differentiation
Q5. For x²+y²=25, find dy/dx where y≠0.
- A. x/y
- B. −y/x
- C. −x/y
- D. 2x+2y
Show correct answer and step-by-step explanation
Correct answer: C. −x/y
Differentiate to get 2x+2yy’=0, then solve for y’.
Applications
Q6. If f'(x)>0 throughout an interval, what is f on that interval?
- A. Decreasing
- B. Constant
- C. Undefined
- D. Increasing
Show correct answer and step-by-step explanation
Correct answer: D. Increasing
A positive derivative gives strict increase.
Second Derivative
Q7. For f(x)=x³, find f”(x).
- A. 6x
- B. 3x²
- C. x²
- D. 6
Show correct answer and step-by-step explanation
Correct answer: A. 6x
Differentiate twice: 3x², then 6x.
Integration
Q8. Evaluate ∫cos x dx.
- A. −sin x+C
- B. sin x+C
- C. tan x+C
- D. −cos x+C
Show correct answer and step-by-step explanation
Correct answer: B. sin x+C
The derivative of sin x is cos x.
Improper Integrals
Q9. Does ∫₁∞ 1/x² dx converge?
- A. No
- B. Only conditionally
- C. Yes, to 1
- D. Yes, to infinity
Show correct answer and step-by-step explanation
Correct answer: C. Yes, to 1
Integrate to 1−1/R then let R tend to infinity.
3D Geometry
Q10. Which is a normal vector to the plane 2x−3y+z=5?
- A. (1,1,1)
- B. (2,3,1)
- C. (5,0,0)
- D. (2,−3,1)
Show correct answer and step-by-step explanation
Correct answer: D. (2,−3,1)
Coefficients of x,y,z form a plane normal.
Solved Examples with Step-by-Step Answers
Example 1: Tangent slope
Problem: Find the tangent slope to y=x²+3x at x=2.
- Differentiate y’=2x+3.
- Substitute x=2 to obtain 4+3.
Final answer: The slope equals 7.
Example 2: Improper integral
Problem: Evaluate ∫₁∞ x⁻² dx.
- Evaluate from 1 to R as [−1/x]₁ᴿ=1−1/R.
- Take R→∞.
Final answer: The integral converges to 1.
Additional Topic-Wise MCQs — Questions 11–20
Additional concept and calculation questions. Select a question’s expandable answer to review the reasoning.
Continuity
Q11. Which function is continuous at every real number?
- A. f(x)=1/x
- B. f(x)=x²+1
- C. f(x)=1/(x−2)
- D. f(x)=tan x
Show correct answer and explanation
Correct answer: B. f(x)=x²+1
Every polynomial is continuous on the real line.
Product Rule
Q12. Differentiate x eˣ.
- A. eˣ
- B. xeˣ
- C. (x+1)eˣ
- D. (x−1)eˣ
Show correct answer and explanation
Correct answer: C. (x+1)eˣ
The product rule gives eˣ+xeˣ.
Quotient Rule
Q13. Differentiate 1/x for x≠0.
- A. 1/x²
- B. −1/x²
- C. −x²
- D. ln|x|
Show correct answer and explanation
Correct answer: B. −1/x²
Write x⁻¹ and apply the power rule.
Tangent Line
Q14. For y=x², which line is tangent at x=1?
- A. y=2x−1
- B. y=x−1
- C. y=2x+1
- D. y=x²+1
Show correct answer and explanation
Correct answer: A. y=2x−1
Point is (1,1) and slope is 2, so y−1=2(x−1).
Optimization
Q15. What is the minimum value of x²−4x+7 over real x?
- A. −3
- B. 2
- C. 3
- D. 7
Show correct answer and explanation
Correct answer: C. 3
Complete the square: (x−2)²+3, minimized at x=2.
Integration Substitution
Q16. Evaluate ∫2x cos(x²)dx.
- A. cos(x²)+C
- B. 2sin x+C
- C. x² sin(x²)+C
- D. sin(x²)+C
Show correct answer and explanation
Correct answer: D. sin(x²)+C
Substitute u=x², du=2x dx.
Area
Q17. What is the area under y=x from x=0 to x=2?
- A. 1
- B. 2
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: B. 2
Integral of x from 0 to 2 is [x²/2]₀²=2.
Improper Integrals
Q18. For p>1, what is ∫₁∞ x⁻ᵖ dx?
- A. 1/(p−1)
- B. 1/(1−p)
- C. p−1
- D. ∞
Show correct answer and explanation
Correct answer: A. 1/(p−1)
Integrate and take the limit at infinity; the result is 1/(p−1).
Three-Dimensional Lines
Q19. Which vector is parallel to the line (x,y,z)=(1,2,3)+t(2,−1,4)?
- A. (1,2,3)
- B. (0,0,1)
- C. (2,−1,4)
- D. (2,1,4)
Show correct answer and explanation
Correct answer: C. (2,−1,4)
The coefficient vector of t is the direction vector.
Mean Value Theorem
Q20. For f(x)=x² on [1,3], which c satisfies f'(c)=[f(3)−f(1)]/(3−1)?
- A. 1
- B. 2
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: B. 2
Average slope is (9−1)/2=4, so 2c=4 and c=2.
Additional Practice MCQs — Questions 21–30
Further foundational and intermediate practice, with four answer options and explanations.
Limits
Q21. Find lim(x→0) (1−cos x)/x² in radians.
- A. 0
- B. 1/2
- C. 1
- D. 2
Show correct answer and explanation
Correct answer: B. 1/2
The standard trigonometric limit is 1/2.
Derivatives
Q22. Differentiate arctan x.
- A. 1/(1+x²)
- B. 1/(1−x²)
- C. −1/(1+x²)
- D. arctan x
Show correct answer and explanation
Correct answer: A. 1/(1+x²)
Derivative of arctan x equals 1/(1+x²).
Chain Rule
Q23. Differentiate ln(x²+1).
- A. 1/(x²+1)
- B. 2x
- C. 2x/(x²+1)
- D. x²+1
Show correct answer and explanation
Correct answer: C. 2x/(x²+1)
Apply the logarithm derivative and the chain rule.
Mean Value Theorem
Q24. What does the Mean Value Theorem guarantee under its hypotheses?
- A. f'(c)=0 always
- B. f(a)=f(b)
- C. f is linear
- D. f'(c) equals the average slope for some c
Show correct 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).
Critical Points
Q25. Where does the derivative of x³−3x vanish?
- A. At x=0
- B. At x=−1 and 1
- C. At x=−3 and 3
- D. At x=1 only
Show correct answer and explanation
Correct answer: B. At x=−1 and 1
3x²−3=0 gives x²=1.
Antiderivatives
Q26. Find ∫sec²x dx.
- A. tan x+C
- B. sec x+C
- C. sin x+C
- D. −cot x+C
Show correct answer and explanation
Correct answer: A. tan x+C
Derivative of tangent is secant squared.
Area
Q27. Find the area between y=2 and y=x on [0,2].
- A. 1
- B. 4
- C. 2
- D. 8
Show correct answer and explanation
Correct answer: C. 2
∫₀²(2−x)dx=2.
Improper Integrals
Q28. For what values of p does ∫₁∞ x^(−p)dx converge?
- A. p<0
- B. p=0
- C. p≤1
- D. p>1
Show correct answer and explanation
Correct answer: D. p>1
The integral converges exactly when p>1.
Planes
Q29. Find the plane through (0,0,3) with normal (1,2,−1).
- A. x+2y−z=3
- B. x+2y−z=−3
- C. x−2y+z=3
- D. x+y+z=3
Show correct answer and explanation
Correct answer: B. x+2y−z=−3
The normal equation gives x+2y−(z−3)=0.
Tangents
Q30. What is the slope of y=sin x at x=0?
- A. 1
- B. 0
- C. −1
- D. π
Show correct answer and explanation
Correct answer: A. 1
The derivative is cos x, giving cos 0=1.
Additional Step-by-Step Solved Example
Optimization
Problem: A rectangle has perimeter 20. What is its greatest possible area?
- Let the side lengths be x and 10−x.
- Write area A(x)=10x−x².
- Solve A'(x)=10−2x=0 to obtain x=5; A”(x)=−2<0.
Final answer: The maximum area is 25 square units.
Additional Practice MCQs — Questions 31–150
Q31. Evaluate lim(x→1) (2x+3).
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
Polynomials are continuous, so substitute x=1 to obtain 5.
Q32. Evaluate lim(x→2) (3x+4).
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
Polynomials are continuous, so substitute x=2 to obtain 10.
Q33. Evaluate lim(x→3) (4x+5).
- A. 20
- B. 17
- C. 18
- D. 16
Show correct answer and explanation
Correct answer: B. 17
Polynomials are continuous, so substitute x=3 to obtain 17.
Q34. Evaluate lim(x→4) (5x+6).
- A. 25
- B. 29
- C. 26
- D. 27
Show correct answer and explanation
Correct answer: C. 26
Polynomials are continuous, so substitute x=4 to obtain 26.
Q35. Evaluate lim(x→5) (6x+7).
- A. 38
- B. 36
- C. 40
- D. 37
Show correct answer and explanation
Correct answer: D. 37
Polynomials are continuous, so substitute x=5 to obtain 37.
Q36. Evaluate lim(x→6) (7x+8).
- A. 50
- B. 51
- C. 49
- D. 53
Show correct answer and explanation
Correct answer: A. 50
Polynomials are continuous, so substitute x=6 to obtain 50.
Q37. Evaluate lim(x→7) (8x+9).
- A. 68
- B. 65
- C. 66
- D. 64
Show correct answer and explanation
Correct answer: B. 65
Polynomials are continuous, so substitute x=7 to obtain 65.
Q38. Evaluate lim(x→8) (9x+10).
- A. 81
- B. 85
- C. 82
- D. 83
Show correct answer and explanation
Correct answer: C. 82
Polynomials are continuous, so substitute x=8 to obtain 82.
Q39. Evaluate lim(x→9) (10x+11).
- A. 102
- B. 100
- C. 104
- D. 101
Show correct answer and explanation
Correct answer: D. 101
Polynomials are continuous, so substitute x=9 to obtain 101.
Q40. Evaluate lim(x→10) (11x+12).
- A. 122
- B. 123
- C. 121
- D. 125
Show correct answer and explanation
Correct answer: A. 122
Polynomials are continuous, so substitute x=10 to obtain 122.
Q41. Find the derivative at x=1 of f(x)=2x².
- A. 7
- B. 4
- C. 5
- D. 3
Show correct answer and explanation
Correct answer: B. 4
f'(x)=4x, so f'(1)=4.
Q42. Find the derivative at x=2 of f(x)=3x².
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
f'(x)=6x, so f'(2)=12.
Q43. Find the derivative at x=3 of f(x)=4x².
- A. 25
- B. 23
- C. 27
- D. 24
Show correct answer and explanation
Correct answer: D. 24
f'(x)=8x, so f'(3)=24.
Q44. Find the derivative at x=4 of f(x)=5x².
- A. 40
- B. 41
- C. 39
- D. 43
Show correct answer and explanation
Correct answer: A. 40
f'(x)=10x, so f'(4)=40.
Q45. Find the derivative at x=5 of f(x)=6x².
- A. 63
- B. 60
- C. 61
- D. 59
Show correct answer and explanation
Correct answer: B. 60
f'(x)=12x, so f'(5)=60.
Q46. Find the derivative at x=6 of f(x)=7x².
- A. 83
- B. 87
- C. 84
- D. 85
Show correct answer and explanation
Correct answer: C. 84
f'(x)=14x, so f'(6)=84.
Q47. Find the derivative at x=7 of f(x)=8x².
- A. 113
- B. 111
- C. 115
- D. 112
Show correct answer and explanation
Correct answer: D. 112
f'(x)=16x, so f'(7)=112.
Q48. Find the derivative at x=8 of f(x)=9x².
- A. 144
- B. 145
- C. 143
- D. 147
Show correct answer and explanation
Correct answer: A. 144
f'(x)=18x, so f'(8)=144.
Q49. Find the derivative at x=9 of f(x)=10x².
- A. 183
- B. 180
- C. 181
- D. 179
Show correct answer and explanation
Correct answer: B. 180
f'(x)=20x, so f'(9)=180.
Q50. Find the derivative at x=10 of f(x)=11x².
- A. 219
- B. 223
- C. 220
- D. 221
Show correct answer and explanation
Correct answer: C. 220
f'(x)=22x, so f'(10)=220.
Q51. Compute ∫₀¹ 2x^1 dx.
- A. 2
- B. 0
- C. 4
- D. 1
Show correct answer and explanation
Correct answer: D. 1
Using the power rule, the definite integral equals 2/(2)=1.
Q52. Compute ∫₀¹ 3x^2 dx.
- A. 1
- B. 2
- C. 0
- D. 4
Show correct answer and explanation
Correct answer: A. 1
Using the power rule, the definite integral equals 3/(3)=1.
Q53. Compute ∫₀¹ 4x^3 dx.
- A. 4
- B. 1
- C. 2
- D. 0
Show correct answer and explanation
Correct answer: B. 1
Using the power rule, the definite integral equals 4/(4)=1.
Q54. Compute ∫₀¹ 5x^4 dx.
- A. 0
- B. 4
- C. 1
- D. 2
Show correct answer and explanation
Correct answer: C. 1
Using the power rule, the definite integral equals 5/(5)=1.
Q55. Compute ∫₀¹ 6x^5 dx.
- A. 2
- B. 0
- C. 4
- D. 1
Show correct answer and explanation
Correct answer: D. 1
Using the power rule, the definite integral equals 6/(6)=1.
Q56. Compute ∫₀¹ 7x^6 dx.
- A. 1
- B. 2
- C. 0
- D. 4
Show correct answer and explanation
Correct answer: A. 1
Using the power rule, the definite integral equals 7/(7)=1.
Q57. Compute ∫₀¹ 8x^7 dx.
- A. 4
- B. 1
- C. 2
- D. 0
Show correct answer and explanation
Correct answer: B. 1
Using the power rule, the definite integral equals 8/(8)=1.
Q58. Compute ∫₀¹ 9x^8 dx.
- A. 0
- B. 4
- C. 1
- D. 2
Show correct answer and explanation
Correct answer: C. 1
Using the power rule, the definite integral equals 9/(9)=1.
Q59. Compute ∫₀¹ 10x^9 dx.
- A. 2
- B. 0
- C. 4
- D. 1
Show correct answer and explanation
Correct answer: D. 1
Using the power rule, the definite integral equals 10/(10)=1.
Q60. Compute ∫₀¹ 11x^10 dx.
- A. 1
- B. 2
- C. 0
- D. 4
Show correct answer and explanation
Correct answer: A. 1
Using the power rule, the definite integral equals 11/(11)=1.
Q61. For f(x)=2x+3, what is the slope of its graph?
- A. 5
- B. 2
- C. 3
- D. 1
Show correct answer and explanation
Correct answer: B. 2
In y=mx+b, the coefficient of x is the slope m=2.
Q62. For f(x)=3x+4, what is the slope of its graph?
- A. 2
- B. 6
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: C. 3
In y=mx+b, the coefficient of x is the slope m=3.
Q63. For f(x)=4x+5, what is the slope of its graph?
- A. 5
- B. 3
- C. 7
- D. 4
Show correct answer and explanation
Correct answer: D. 4
In y=mx+b, the coefficient of x is the slope m=4.
Q64. For f(x)=5x+6, what is the slope of its graph?
- A. 5
- B. 6
- C. 4
- D. 8
Show correct answer and explanation
Correct answer: A. 5
In y=mx+b, the coefficient of x is the slope m=5.
Q65. For f(x)=6x+7, what is the slope of its graph?
- A. 9
- B. 6
- C. 7
- D. 5
Show correct answer and explanation
Correct answer: B. 6
In y=mx+b, the coefficient of x is the slope m=6.
Q66. For f(x)=7x+8, what is the slope of its graph?
- A. 6
- B. 10
- C. 7
- D. 8
Show correct answer and explanation
Correct answer: C. 7
In y=mx+b, the coefficient of x is the slope m=7.
Q67. For f(x)=8x+9, what is the slope of its graph?
- A. 9
- B. 7
- C. 11
- D. 8
Show correct answer and explanation
Correct answer: D. 8
In y=mx+b, the coefficient of x is the slope m=8.
Q68. For f(x)=9x+10, what is the slope of its graph?
- A. 9
- B. 10
- C. 8
- D. 12
Show correct answer and explanation
Correct answer: A. 9
In y=mx+b, the coefficient of x is the slope m=9.
Q69. For f(x)=10x+11, what is the slope of its graph?
- A. 13
- B. 10
- C. 11
- D. 9
Show correct answer and explanation
Correct answer: B. 10
In y=mx+b, the coefficient of x is the slope m=10.
Q70. For f(x)=11x+12, what is the slope of its graph?
- A. 10
- B. 14
- C. 11
- D. 12
Show correct answer and explanation
Correct answer: C. 11
In y=mx+b, the coefficient of x is the slope m=11.
Q71. Find the radius of the circle (x−1)²+(y+2)²=4.
- A. 3
- B. 1
- C. 5
- D. 2
Show correct answer and explanation
Correct answer: D. 2
The right-hand side is r²=4; hence radius r=2.
Q72. Find the radius of the circle (x−2)²+(y+3)²=9.
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
The right-hand side is r²=9; hence radius r=3.
Q73. Find the radius of the circle (x−3)²+(y+4)²=16.
- A. 7
- B. 4
- C. 5
- D. 3
Show correct answer and explanation
Correct answer: B. 4
The right-hand side is r²=16; hence radius r=4.
Q74. Find the radius of the circle (x−4)²+(y+5)²=25.
- A. 4
- B. 8
- C. 5
- D. 6
Show correct answer and explanation
Correct answer: C. 5
The right-hand side is r²=25; hence radius r=5.
Q75. Find the radius of the circle (x−5)²+(y+6)²=36.
- A. 7
- B. 5
- C. 9
- D. 6
Show correct answer and explanation
Correct answer: D. 6
The right-hand side is r²=36; hence radius r=6.
Q76. Find the radius of the circle (x−6)²+(y+7)²=49.
- A. 7
- B. 8
- C. 6
- D. 10
Show correct answer and explanation
Correct answer: A. 7
The right-hand side is r²=49; hence radius r=7.
Q77. Find the radius of the circle (x−7)²+(y+8)²=64.
- A. 11
- B. 8
- C. 9
- D. 7
Show correct answer and explanation
Correct answer: B. 8
The right-hand side is r²=64; hence radius r=8.
Q78. Find the radius of the circle (x−8)²+(y+9)²=81.
- A. 8
- B. 12
- C. 9
- D. 10
Show correct answer and explanation
Correct answer: C. 9
The right-hand side is r²=81; hence radius r=9.
Q79. Find the radius of the circle (x−9)²+(y+10)²=100.
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
The right-hand side is r²=100; hence radius r=10.
Q80. Find the radius of the circle (x−10)²+(y+11)²=121.
- A. 11
- B. 12
- C. 10
- D. 14
Show correct answer and explanation
Correct answer: A. 11
The right-hand side is r²=121; hence radius r=11.
Q81. Evaluate the second derivative of f(x)=2x³ at x=1.
- A. 15
- B. 12
- C. 13
- D. 11
Show correct answer and explanation
Correct answer: B. 12
Differentiate twice: f”(x)=12x; at 1 this is 12.
Q82. Evaluate the second derivative of f(x)=3x³ at x=2.
- A. 35
- B. 39
- C. 36
- D. 37
Show correct answer and explanation
Correct answer: C. 36
Differentiate twice: f”(x)=18x; at 2 this is 36.
Q83. Evaluate the second derivative of f(x)=4x³ at x=3.
- A. 73
- B. 71
- C. 75
- D. 72
Show correct answer and explanation
Correct answer: D. 72
Differentiate twice: f”(x)=24x; at 3 this is 72.
Q84. Evaluate the second derivative of f(x)=5x³ at x=4.
- A. 120
- B. 121
- C. 119
- D. 123
Show correct answer and explanation
Correct answer: A. 120
Differentiate twice: f”(x)=30x; at 4 this is 120.
Q85. Evaluate the second derivative of f(x)=6x³ at x=5.
- A. 183
- B. 180
- C. 181
- D. 179
Show correct answer and explanation
Correct answer: B. 180
Differentiate twice: f”(x)=36x; at 5 this is 180.
Q86. Evaluate the second derivative of f(x)=7x³ at x=6.
- A. 251
- B. 255
- C. 252
- D. 253
Show correct answer and explanation
Correct answer: C. 252
Differentiate twice: f”(x)=42x; at 6 this is 252.
Q87. Evaluate the second derivative of f(x)=8x³ at x=7.
- A. 337
- B. 335
- C. 339
- D. 336
Show correct answer and explanation
Correct answer: D. 336
Differentiate twice: f”(x)=48x; at 7 this is 336.
Q88. Evaluate the second derivative of f(x)=9x³ at x=8.
- A. 432
- B. 433
- C. 431
- D. 435
Show correct answer and explanation
Correct answer: A. 432
Differentiate twice: f”(x)=54x; at 8 this is 432.
Q89. Evaluate the second derivative of f(x)=10x³ at x=9.
- A. 543
- B. 540
- C. 541
- D. 539
Show correct answer and explanation
Correct answer: B. 540
Differentiate twice: f”(x)=60x; at 9 this is 540.
Q90. Evaluate the second derivative of f(x)=11x³ at x=10.
- A. 659
- B. 663
- C. 660
- D. 661
Show correct answer and explanation
Correct answer: C. 660
Differentiate twice: f”(x)=66x; at 10 this is 660.
Q91. Find ∫₀^2 2x dx.
- A. 5
- B. 3
- C. 7
- D. 4
Show correct answer and explanation
Correct answer: D. 4
The antiderivative is x²; evaluating at 2 and 0 gives 4.
Q92. Find ∫₀^3 2x dx.
- A. 9
- B. 10
- C. 8
- D. 12
Show correct answer and explanation
Correct answer: A. 9
The antiderivative is x²; evaluating at 3 and 0 gives 9.
Q93. Find ∫₀^4 2x dx.
- A. 19
- B. 16
- C. 17
- D. 15
Show correct answer and explanation
Correct answer: B. 16
The antiderivative is x²; evaluating at 4 and 0 gives 16.
Q94. Find ∫₀^5 2x dx.
- A. 24
- B. 28
- C. 25
- D. 26
Show correct answer and explanation
Correct answer: C. 25
The antiderivative is x²; evaluating at 5 and 0 gives 25.
Q95. Find ∫₀^6 2x dx.
- A. 37
- B. 35
- C. 39
- D. 36
Show correct answer and explanation
Correct answer: D. 36
The antiderivative is x²; evaluating at 6 and 0 gives 36.
Q96. Find ∫₀^7 2x dx.
- A. 49
- B. 50
- C. 48
- D. 52
Show correct answer and explanation
Correct answer: A. 49
The antiderivative is x²; evaluating at 7 and 0 gives 49.
Q97. Find ∫₀^8 2x dx.
- A. 67
- B. 64
- C. 65
- D. 63
Show correct answer and explanation
Correct answer: B. 64
The antiderivative is x²; evaluating at 8 and 0 gives 64.
Q98. Find ∫₀^9 2x dx.
- A. 80
- B. 84
- C. 81
- D. 82
Show correct answer and explanation
Correct answer: C. 81
The antiderivative is x²; evaluating at 9 and 0 gives 81.
Q99. Find ∫₀^10 2x dx.
- A. 101
- B. 99
- C. 103
- D. 100
Show correct answer and explanation
Correct answer: D. 100
The antiderivative is x²; evaluating at 10 and 0 gives 100.
Q100. Find ∫₀^11 2x dx.
- A. 121
- B. 122
- C. 120
- D. 124
Show correct answer and explanation
Correct answer: A. 121
The antiderivative is x²; evaluating at 11 and 0 gives 121.
Q101. Solve 2x=6 for x.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
Divide both sides by 2 to obtain x=3.
Q102. Solve 3x=12 for x.
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
Divide both sides by 3 to obtain x=4.
Q103. Solve 4x=20 for x.
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
Divide both sides by 4 to obtain x=5.
Q104. Solve 5x=30 for x.
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
Divide both sides by 5 to obtain x=6.
Q105. Solve 6x=42 for x.
- A. 10
- B. 7
- C. 8
- D. 6
Show correct answer and explanation
Correct answer: B. 7
Divide both sides by 6 to obtain x=7.
Q106. Solve 7x=56 for x.
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
Divide both sides by 7 to obtain x=8.
Q107. Solve 8x=72 for x.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
Divide both sides by 8 to obtain x=9.
Q108. Solve 9x=90 for x.
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
Divide both sides by 9 to obtain x=10.
Q109. Solve 10x=110 for x.
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
Divide both sides by 10 to obtain x=11.
Q110. Solve 11x=132 for x.
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
Divide both sides by 11 to obtain x=12.
Q111. Find the x-intercept of y=2x−6.
- A. 4
- B. 2
- C. 6
- D. 3
Show correct answer and explanation
Correct answer: D. 3
Set y=0 and solve 2x=6, giving x=3.
Q112. Find the x-intercept of y=3x−12.
- A. 4
- B. 5
- C. 3
- D. 7
Show correct answer and explanation
Correct answer: A. 4
Set y=0 and solve 3x=12, giving x=4.
Q113. Find the x-intercept of y=4x−20.
- A. 8
- B. 5
- C. 6
- D. 4
Show correct answer and explanation
Correct answer: B. 5
Set y=0 and solve 4x=20, giving x=5.
Q114. Find the x-intercept of y=5x−30.
- A. 5
- B. 9
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: C. 6
Set y=0 and solve 5x=30, giving x=6.
Q115. Find the x-intercept of y=6x−42.
- A. 8
- B. 6
- C. 10
- D. 7
Show correct answer and explanation
Correct answer: D. 7
Set y=0 and solve 6x=42, giving x=7.
Q116. Find the x-intercept of y=7x−56.
- A. 8
- B. 9
- C. 7
- D. 11
Show correct answer and explanation
Correct answer: A. 8
Set y=0 and solve 7x=56, giving x=8.
Q117. Find the x-intercept of y=8x−72.
- A. 12
- B. 9
- C. 10
- D. 8
Show correct answer and explanation
Correct answer: B. 9
Set y=0 and solve 8x=72, giving x=9.
Q118. Find the x-intercept of y=9x−90.
- A. 9
- B. 13
- C. 10
- D. 11
Show correct answer and explanation
Correct answer: C. 10
Set y=0 and solve 9x=90, giving x=10.
Q119. Find the x-intercept of y=10x−110.
- A. 12
- B. 10
- C. 14
- D. 11
Show correct answer and explanation
Correct answer: D. 11
Set y=0 and solve 10x=110, giving x=11.
Q120. Find the x-intercept of y=11x−132.
- A. 12
- B. 13
- C. 11
- D. 15
Show correct answer and explanation
Correct answer: A. 12
Set y=0 and solve 11x=132, giving x=12.
Q121. Find the area of a triangle with base 2 and height 3.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
Area=(1/2)×base×height=3.
Q122. Find the area of a triangle with base 3 and height 4.
- A. 5
- B. 9
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: C. 6
Area=(1/2)×base×height=6.
Q123. Find the area of a triangle with base 4 and height 5.
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
Area=(1/2)×base×height=10.
Q124. Find the area of a triangle with base 5 and height 6.
- A. 15
- B. 16
- C. 14
- D. 18
Show correct answer and explanation
Correct answer: A. 15
Area=(1/2)×base×height=15.
Q125. Find the area of a triangle with base 6 and height 7.
- A. 24
- B. 21
- C. 22
- D. 20
Show correct answer and explanation
Correct answer: B. 21
Area=(1/2)×base×height=21.
Q126. Find the area of a triangle with base 7 and height 8.
- A. 27
- B. 31
- C. 28
- D. 29
Show correct answer and explanation
Correct answer: C. 28
Area=(1/2)×base×height=28.
Q127. Find the area of a triangle with base 8 and height 9.
- A. 37
- B. 35
- C. 39
- D. 36
Show correct answer and explanation
Correct answer: D. 36
Area=(1/2)×base×height=36.
Q128. Find the area of a triangle with base 9 and height 10.
- A. 45
- B. 46
- C. 44
- D. 48
Show correct answer and explanation
Correct answer: A. 45
Area=(1/2)×base×height=45.
Q129. Find the area of a triangle with base 10 and height 11.
- A. 58
- B. 55
- C. 56
- D. 54
Show correct answer and explanation
Correct answer: B. 55
Area=(1/2)×base×height=55.
Q130. Find the area of a triangle with base 11 and height 12.
- A. 65
- B. 69
- C. 66
- D. 67
Show correct answer and explanation
Correct answer: C. 66
Area=(1/2)×base×height=66.
Q131. Evaluate the derivative of f(x)=x^2 at x=1.
- A. 3
- B. 1
- C. 5
- D. 2
Show correct answer and explanation
Correct answer: D. 2
By the power rule, f'(x)=2x^1, hence f'(1)=2.
Q132. Evaluate the derivative of f(x)=x^3 at x=1.
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
By the power rule, f'(x)=3x^2, hence f'(1)=3.
Q133. Evaluate the derivative of f(x)=x^4 at x=1.
- A. 7
- B. 4
- C. 5
- D. 3
Show correct answer and explanation
Correct answer: B. 4
By the power rule, f'(x)=4x^3, hence f'(1)=4.
Q134. Evaluate the derivative of f(x)=x^5 at x=1.
- A. 4
- B. 8
- C. 5
- D. 6
Show correct answer and explanation
Correct answer: C. 5
By the power rule, f'(x)=5x^4, hence f'(1)=5.
Q135. Evaluate the derivative of f(x)=x^6 at x=1.
- A. 7
- B. 5
- C. 9
- D. 6
Show correct answer and explanation
Correct answer: D. 6
By the power rule, f'(x)=6x^5, hence f'(1)=6.
Q136. Evaluate the derivative of f(x)=x^7 at x=1.
- A. 7
- B. 8
- C. 6
- D. 10
Show correct answer and explanation
Correct answer: A. 7
By the power rule, f'(x)=7x^6, hence f'(1)=7.
Q137. Evaluate the derivative of f(x)=x^8 at x=1.
- A. 11
- B. 8
- C. 9
- D. 7
Show correct answer and explanation
Correct answer: B. 8
By the power rule, f'(x)=8x^7, hence f'(1)=8.
Q138. Evaluate the derivative of f(x)=x^9 at x=1.
- A. 8
- B. 12
- C. 9
- D. 10
Show correct answer and explanation
Correct answer: C. 9
By the power rule, f'(x)=9x^8, hence f'(1)=9.
Q139. Evaluate the derivative of f(x)=x^10 at x=1.
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
By the power rule, f'(x)=10x^9, hence f'(1)=10.
Q140. Evaluate the derivative of f(x)=x^11 at x=1.
- A. 11
- B. 12
- C. 10
- D. 14
Show correct answer and explanation
Correct answer: A. 11
By the power rule, f'(x)=11x^10, hence f'(1)=11.
Q141. For f(x)=(x−2)²+3, find the minimum value on R.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
A square is nonnegative and vanishes at x=2; the minimum is 3.
Q142. For f(x)=(x−3)²+4, find the minimum value on R.
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
A square is nonnegative and vanishes at x=3; the minimum is 4.
Q143. For f(x)=(x−4)²+5, find the minimum value on R.
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
A square is nonnegative and vanishes at x=4; the minimum is 5.
Q144. For f(x)=(x−5)²+6, find the minimum value on R.
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
A square is nonnegative and vanishes at x=5; the minimum is 6.
Q145. For f(x)=(x−6)²+7, find the minimum value on R.
- A. 10
- B. 7
- C. 8
- D. 6
Show correct answer and explanation
Correct answer: B. 7
A square is nonnegative and vanishes at x=6; the minimum is 7.
Q146. For f(x)=(x−7)²+8, find the minimum value on R.
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
A square is nonnegative and vanishes at x=7; the minimum is 8.
Q147. For f(x)=(x−8)²+9, find the minimum value on R.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
A square is nonnegative and vanishes at x=8; the minimum is 9.
Q148. For f(x)=(x−9)²+10, find the minimum value on R.
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
A square is nonnegative and vanishes at x=9; the minimum is 10.
Q149. For f(x)=(x−10)²+11, find the minimum value on R.
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
A square is nonnegative and vanishes at x=10; the minimum is 11.
Q150. For f(x)=(x−11)²+12, find the minimum value on R.
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
A square is nonnegative and vanishes at x=11; the minimum is 12.
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