varsity math questions-calculus

Varsity Calculus Questions

Q: Evaluate the integral of 3 x squared – 4 d x from zero to 6.

A: 192

Q: Evaluate the integral of 3 x squared – 4 d x from zero to 8.

A: 480

Q: Evaluate the integral of 6 x squared + 8 from -3 to 2.

A: 110

Q: Evaluate the integral of cosine x d x from pi over 4 to pi.

A: negative the square root of 2 over 2

Q: Evaluate the integral of 2 x squared +2 x - 3 d x from zero to 2. Express your answer as a mixed number.

A: 3 1/3

Q: Evaluate the integral of x squared from -2 to 3.

A: 35/3 (or 11 2/3)

Q: Evaluate the integral of 6 x squared – 4x from -3 to 2.

A: 80

Q: Evaluate the integral of the cosine x dx from pi over 6 to pi over 4?

A: square root of 2 minus 1 all over 2

Q: Evaluate the integral of the negative sine x dx from pi over 4 to pi.

A: negative 1 minus the square root of 2 over 2 or negative 2 minus the square root of 2 all over 2

Q: Evaluate the integral of x times the quantity (x squared + 1), quantity to the 4th dx from -1 to 1.

A: 0

Q: What is the slope of the tangent line to y = x cubed – 12 x squared + 4 x + 4 at x = 4?

A: 148

Q: What is the area between the curves y = x squared + 2 and y = -x from x= 0 to x = 1?

A: 17/6 or 2 5/6

Q: How many critical points does the following function have: y = x cubed + 10 x - 1?

A: 0

Q: Determine the maximum value of the curve f of x equals -6 x squared + 8 x – 9.

A: -19/3 or -6 1/3

Q: What is the minimum point of the curve y = 5x squared plus 20 x - 13? Express the answer as an ordered pair.

A: (-2, -33)

Q: Find the minimum point of the graph of the equation y = 4x squared - 16 x + 5. Express the answer as an ordered pair using improper fractions if necessary.

A: (2, -11)

Q: Find the minimum point of the graph of the equation y = 2 x squared - 16 x + 13. Express the answer as an ordered pair using improper fractions if necessary.

A: (4, -19)

Q: Find the minimum point of the graph of the equation y = 2x squared - 20 x + 15. Express the answer as an ordered pair using improper fractions if necessary.

A: (5, -35)

Q: Evaluate the limit as y approaches 4 of the quantity (y - 4), close quantity, over the quantity (y squared – 5 y + 4).

A: 1/3

Q: Evaluate the limit as x approaches 10 of x cubed over the quantity x squared + 900?

A: 1

Q: Evaluate the limit as x approaches infinity of x times the quantity (x – 2), all over 4x squared?

A: 1/4

Q: Evaluate the limit as x approaches infinity of x times the quantity (x - 6 over x), close quantity all over the quantity 3 x squared + 9.

A: 1/3

Q: Determine the limit as x approaches zero of the cube root of the quantity x cubed plus 8, close quantity, minus 2, all over x cubed.

A: 1/12

Q: What is the limit as x approaches infinity of the function f of x equals the quantity x cubed plus 3 over the quantity x cubed plus x?

A: 1

Q: Determine the limit of sine x over x as x approaches pi.

A: 0

Q: Determine the limit as x approaches zero of sine 7 x over sine 3 x.

A: 7/3

Q: Determine the limit as x approaches infinity of tangent x times e to the x all over x to the x.

A: 0

Q: Determine the limit as x approaches 8 of the quantity (x squared – 4), close quantity divided by the quantity (x squared + x – 6) close quantity. Express your answer as a fraction, if necessary.

A: 10/11

Q: Determine the limit as x approaches zero of tangent of 4x over the quantity x – 5 times the square root of 2.

A: 0

Q: Determine the limit of the absolute value as x approaches infinity of x divided by the quantity 1 plus x squared.

A: 0

Q: Determine the limit as x approaches zero of tangent of 2x over the sine of 2x.

A: 1

Q: Given: f of x = x squared + the quantity 3 over x squared, find f prime of x. Express all exponents as positive numbers?

A: 2x – 6 over x cubed

Q: Given: f of x = 15 x squared times 18 over x cubed, find f prime of x. Express all exponents as positive numbers.

A: - 270 over x squared

Q: Given: f of x = 10 x squared times 18 over x cubed, find f prime of x. Express all exponents as positive numbers.

A: - 180 over x squared

Q: Given: f of x = 16 over x to the 6th times 4 x squared, find f prime of x. Express all exponents as positive numbers?

A: -256 over x to the 5th

Q: What is the slope of the tangent line to y = x cubed – 8 x squared – 4 x + 9 at x =2?

A: -24

Q: Find the f quadruple prime of x if f of x equals x cubed + 13 x squared.

A: 0

Q: Find the f double prime of x if f of x equals sine x.

A: negative sine of x

Q: Evaluate the derivative of 5 x cubed – 15x for x equals 3.

A: 120

Q: Determine the fourth derivative of y = sin x.

A: sin x

Q: Determine the second derivative of 7/3 x to the 6th power + 2x to the 2/3 power + 15.

A: 70x to the 4th – 4/9x to the -4/3 power or 70x to the 4th – 4 over 9x to the 4/3 power

Q: Given the curve: y = 3 x cubed + 6 x - 18; what is the value of the second derivative at x = 10?

A: 180

Q: Given the curve: y = 3 x cubed + 6 x - 18; what is the value of the second derivative at x = 6?

A: 108

Q: Given the curve: y = x cubed + 6 x squared; what is the value of the second derivative at x = 3?

A: 30

Q: Given the curve: y = 5 x cubed + 2 x - 18; what is the value of the second derivative at x = -3?

A: -90

Q: Given the curve: y = 4x cubed – 10x squared + 16, determine the value of the second derivative for x = -3.

A: -92

Q: Given the curve: y = 4x cubed – 11 x squared + 9 x, determine the value of the second derivative for x = 5.

A: 98

Q: Given the curve: y = 4x cubed – 11x squared + 9x, determine the value of the second derivative for x = -4.

A: -118

Q: Given the curve: y = 2x to the 4th – 5x cubed + 4x squared – 3x + 2, determine the value of its derivative at x = 2.

A: 17

Q: Given f of x = 1 over x cubed, determine f prime of x. Express all exponents as positive numbers.

A: -3 over x to the 4th

Q: What is the value of the derivative of y equals 5 x squared plus 20 x – 13 for x equal to 10?

A: 120

Q: What term applied to an infinite series that does not approach a limit? The opposite case is “convergent”.

A: divergent

Q: Determine the indefinite integrals of the following functions:

1. integral of u dv –“integral of U D V”2. integral of secant squared x dx3. integral of cotangent u du4. integral of u e to the u du

1. uv – the integral of v du2. tangent x + C3. natural log of the absolute value sine u + C4. the quantity (u -1) e to the u power + C

Q: Differentiate the following:

1. the quantity (4x – 2 x squared) times the quantity (2 + 4x)2. sine of 4x3. the square root of sine x4. the quantity (2x – 7) cubed

1. -24 x squared + 24 x + 82. 4 times the cosine of 4x3. ½ cotangent of x times the square root of the sine of x

(or cosine x over 2 times the square root of sine x)4. 6 times the quantity (2x – 7) squared (or 24 x squared – 168 x + 294)

Q: Differentiate the following and simplify completely:

1. x times the quantity 3 x minus 82. the quantity (2x + 3) times the quantity (3x - 2)3. the quantity (x + 1) cubed4. x times the quantity (2x – 1) times the quantity (2x + 1)

1. 6x - 82. 12x + 53. 3 x squared + 6x +34. 12 x squared - 1

Q: Determine the area under the following curves:

1. y = x over 4 from x = 0 to x = 32. y = 6 x squared + 10 from x = 0 to x = 23. y = 1 – absolute value of x from x = -2 to x = 2

1. 9/8 or 1 1/82. 363. 0


1. y = 8 from x = 0 to x = 32. y = 2x + 3 from x = 0 to x = 23. y = 1 – absolute value of x from x = -1 to x = 14. y = the square root of the quantity (9 – x squared) from x = -3 to x = 3

1. 242. 103. 14. 9 pi over 2


1. y = 7 from x = 0 to x = 62. y = 3 x + 5 from x = 0 to x = 63. y = 1 over x squared from x = 1 to x = 44. y = 4 - x squared from x = 1 to x = 4

1. 422. 843. 3/44. -9


1. y = 7 from x = 0 to x = 42. y = 2 x + 5 from x = 0 to x = 43. y = 1 over x squared from x = 1 to x = 34. y = 4 - x squared from x = 0 to x = 2

1. 282. 363. 2/34. 16/3

Q: Given the curve: y = 2 times the quantity (x squared – 9) close quantity all over the quantity (x squared – 4).

1. and 2. What are the equations of the vertical asymptotes?3. What is the equation of the horizontal asymptote?4. How many inflection points are there?

1. x = -22. x = 23. y = 24. 0

Q: Determine the derivatives of the following functions. Factor the solutions when possible.

1. y = the quantity (t squared + 1), close quantity, times the quantity (t cubed + t squared + 1)

2. y = the quantity (1 + sine x) close quantity to the 4th3. y = cosine 2x sine 3x4. y = cosine of the quantity 3 x squared

1. 2t times the quantity (t cubed + t squared + 1) + the quantity (3t squared + 2t) times the quantity (t squared + 1),

2. 4 times cosine x times the quantity (1 + sine x) close quantity, cubed3. 3 cosine 2x cosine 3x - 2 sine 2x cosine 3x4. -6 x times the sine of 3 x squared

Q: Determine the equations of the horizontal asymptotes for the following functions:

1. y equals 5 + e to the -x squared2. y equals the quantity (7 + x) over the quantity (x squared - 9)3. y equals sine squared of 1 over x4. y equals the quantity (x cubed - 9 x + 4) over the quantity (2 x cubed + 6 x + 7)

1. y = 52. y = 03. y = 04. y = 1/2

Q: Given f of x equals x cubed over the quantity (x – 5) squared, evaluate the following limits:

1. The limit as x approaches 02. The limit as x approaches -53. The limits as x approached infinity4. The limit as x approaches 5

1. 02. -5/4 or -1/1/43. no limit or infinity4. no limit or infinity

Q: Evaluate the following limits using limit laws and L’Hopital’s rule:

1. limit as x approaches 3 of the quantity (x + 3) over the quantity (x – 3) squared2. limit as x approaches zero of sine 7x over x3. limit as x approaches zero of sine 5x over sine 2x4. limit as x approaches zero of secant 2x times tangent 2x all over x

1. 02. 73. 5/24. 2

Q: Evaluate the following limits:

1. x approaches -9 of x squared2. x approaches pi over 4 of sine x3. x approaches -1 of the quantity (x + 1), over the quantity (x squared – x – 2)4. limit as x approaches infinity of 15 x cubed over 5 x to the 4th

1. 812. square root of 2 over 23. -1/34. 0

Q: Determine these limits:

1. Limit as n approaches infinity for -1/8 to the nth power.2. Limit as n approaches infinity for (5n + 1) over (10n – 50).3. Limit as n approaches infinity for 1 + the quantity (4n + 3) over (4n – 2).4. Limit as n approaches infinity for 6x over (x squared + 16)

1. 02. 1/23. 24. 0

Q: Given the curve: y = -6x squared + 8x – 9:

1. Find the derivative.2. Find the second derivative.3. Find the third derivative.4. Find the maximum value.

1. -12x + 82. -123. 04. -19/3 or -6 1/3

Q: Given the curve: y = -17 x squared + 9 x - 8.

1. Find the derivative.2. Find the second derivative.3. Find the third derivative.4. Find the maximum value. Express your answers as an improper fraction.

1. -34 x + 92. -343. 04. -463/6845

Q: Given the curve: y = x over the quantity (x-2)

1. How many maxima and minima are there?2. How many inflection points are there?3. What is the equation of the vertical asymptote?4. What is the equation of the horizontal asymptote?

1. 02. 03. x = 24. y = 1

Q: Given the curve: y = x cubed + 10 x - 1

1. What is the value of the derivative for x = 2?2. What is the value of the second derivative for x = 2?3. What are the critical points?4. What is the inflection point?

1. 22 2. 12 3. There are none 4. (0, -1)

Q: Given the curve: y = -3 x to the 5th - 5 x cubed, answer the following.

1. What is the value of the derivative at x = 6?2. What is the value of the second derivative at x = 6?3. How many critical points are there?4. How many inflection points are there?

1. -189002. -127803. 34. 1

Q: Given the curve: y = x to the 4th - 6 x squared - 15.

1. What is the value of the derivative at x=2?2. What is the value of the second derivative at x = 2?3. How many critical points are there?4. How many inflection points are there?

1. 82. 363. 34. 2

Q: Given the curve: y = 2x to the 6th - 24 x squared

1. What is the value of the derivative for x = 2?2. What is the value of the second derivative for x = 2?3. How many critical points are there?4. How many inflection points are there?

1. 288 2. 912 3. 3 4. 2

Q: Given the curve: y = 5x cubed - 14 x squared + 9 x


1. 13 2. 32 3. 2 4. 1

Q: Given the curve: y = 6x to the 4th – 5 x cubed + 4 x squared – 3 x + 2


1. 145 2. 236 3. 3 4. 0

Q: Given the curve: y = 11 x cubed + 4 x squared – 3.

1. The first derivative.2. The second derivative.3. The third derivative.4. The integral of the function.

1. 33 x squared + 8x 2. 66 x + 83. 66 4. 11/4 x to the 4th + 4/3 x cubed – 3 x + C

(or 11x to the 4th over 4 + 4 x cubed over 3 – 3x + C)

Q: Given the curve: y = 8x to the 4th – 12 x cubed + 16 over the interval -2 is less than x which is less than 2. Express answers as improper fractions where needed:

1. What is the first derivative?2. What is the second derivative?3. What is the maximum value of this interval?4. What is the minimum value of this interval?

1. 32 x cubed – 36 x squared2. 96 x squared – 72 x3. 2404. 6005/512

Q: Given the curve: y = 2 x cubed – 6 x squared = 15 x - 9

1. What is the value of the derivative at x = 9?2. What is the value of the second derivative at x = 9?3. What is the value of the third derivative at x = 9?4. What is the value of the fourth derivative at x = 9?

1. 3932. 963. 124. 0

Q: Given the curve: y = 2 x cubed - 6 x squared + 15 x - 9.


1. 105 2. 48 3. 12 4. 0

Q: Given the curve: y = 2 x cubed - 9 x squared + 10 x - 4.


1. 70 2. 42 3. 12 4. 0

Q: Given the curve: y = 11 x cubed + 5 x squared - 3

1. Find the first derivative.2. Find the second derivative.3. Find the third derivative.4. Find the integral of the function.

1. 33 x squared + 10 x2. 66 x + 103. 664. 11/4 x to the 5th + 5/3 x cubed – 3 x + C

Q: Given: f of x equals 16 x to the 4th – 11 x cubed + e to the 4x – cosine x + 100, find:

1. The first derivative.2. The second derivative.3. The third derivative.4. The fourth derivative.

1. 64 x cubed – 33 x squared + 4e to the 4x + sine x2. 192 x squared – 66 x + 16e to the 4x + cosine x3. 384 x – 66 + 64e to the 4x – sine x4. 384 + 256e to the 4x – cosine x

Q: Given: f of x equals 13 x to the 4th – 11 x cubed + e to the 4x – cosine x, find:


1. 52 x cubed – 33 x squared + 4e to the 4x + sine x2. 156 x squared – 66 x + 16e to the 4x + cosine x3. 312 x – 66 + 64e to the 4x – sine x4. 312 + 256e to the 4x – cosine x

Q: Given: f of x equals 13/2 x to the 4th – 14/3 x cubed – 42 x – 111/7, find:


1. 26 x cubed – 14 x squared - 422. 78 x squared – 28 x 3. 156 x – 28 4. 156

Q: Given: f of x = 11/3 x to the 6th + 11 x to the 4th + 24x + 9, find:


1. 22 x to the 5th + 44 x cubed + 242. 110 x to the 4th + 132 x squared3. 440 x cubed + 264 x4. 1320 x squared + 264

Q: Given: f of x = 6 x to the 6th + 19 x to the 4th + 18x + 15, find:



Q: Given: f of x = 7/3 x to the 6th + 11 x to the 4th + 20 x + 8, find:



Q: Given: f of x equals 12 x to the 4th – 4 x cubed + 2 e to the 2x + cosine x, find:


1. 48 x cubed – 12 x squared + 4e to the 2x - sine x2. 144 x squared – 24 x + 8e to the 2x - cosine x3. 288 x – 24 + 16e to the 2x + sine x4. 288 + 32e to the 2x + cosine x

Q: Given: f of x equals 16 x to the 4th – 11 x cubed + e to the 4x - cosine x, find:


1. 64 x cubed – 33 x squared + 4 e to the 4x + sine x2. 192 x squared – 66 x + 16 e to the 4x + cosine x3. 384 x – 66 + 64 e to the 4x - sine x4. 384 + 256 e to the 2x - cosine x

Q: Given: f of x = 9 x to the 10th power + 18 x to the 5th power + 36 x- 72, find:


1. 9 x to the 9th + 90 x to the 4th + 362. 810 x to the 8th + 360 x cubed3. 6480 x to the 7th + 1080 x squared4. 45360 x to the 6th + 2160 x

Q: Given: f of x = x to the 4th + 8x cubed + 21x + 8, find:


1. 4 x cubed + 24 x squared + 212. 12 x squared + 48 x3. 24 x + 484. 24

Q: Given: f of x equals 7 x to the 4th – 14 x cubed + 10 x – 17, find:


1. 28 x cubed – 42 x squared + 102. 84 x squared – 84 x 3. 168 x – 84 4. 168

Q: Given a function and a value of x, determine the value of the derivative of the function for that value of x.

1. y = 3 x cubed + 4 x squared + x + 8; x = 3.2. y = 4 x to the 5th + x squared; x = 2.3. y = x to the 50th; x = -1.4. y = 1/3 cubed + ½ x squared + x + 1; x = 3.14.

1. 1062. 3243. -504. 13.9996

Q: Given the curve: y = 12 x squared – 5 x + 20.

1. What is the second derivative?2. What is the x value of the extreme point?3. How many inflection points does it have?4. What is the minimum value? Express your answer as a mixed number

1. 242. 5/243. 04. 19 23/48

Q: Given the curve: y = x cubed over the quantity (x-1).

1. How many critical points are there?2. How many point of inflection are there?3. How many vertical asymptotes are there?4. How many horizontal asymptotes are there?

1. 32. 23. 24. 0

Q: Given the curve: y = x over the quantity (x-2).

1. How many total maxima and minima are there?2. How many point of inflection are there?3. What is the equation of the vertical asymptote?4. What is the equation of the horizontal asymptote?

1. 02. 03. x = 24. y = 1

varsity math questions-calculus

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