Quiz

Question 1

Find the limit of (2x^3 - 5x^2 + 3x + 1) / (x^2 - 4x + 3) as x approaches 1.

Answer 1

The limit is 1

Question 2

Determine the equation of the tangent line to the curve y = x^3 + 2x^2 - 5x + 1 at the point where x = 2.

Answer 2

The equation of the tangent line is y = 13x - 17.

Question 3

Find the derivative of f(x) = (4x^2 - 3x + 2) / (5x - 1).

Answer 3

The derivative is f’(x) = (20x + 3)/(5x - 1)^2.

Question 4

Evaluate the integral of e^(-x^2) with respect to x.

Answer 4

The integral of e^(-x^2) does not have a standard form in terms of elementary functions but is closely related to the error function, denoted as erf(x).

Question 5

Calculate the second derivative of g(x) = sin(x) / x.

Answer 5

The second derivative is g’‘(x) = (2 - 2sin(x)/x - cos(x)/x^2).

Question 6

Determine the absolute maximum and minimum values of the function f(x) = x^3 - 6x^2 + 9x + 2 on the interval [0, 4].

Answer 6

The absolute maximum is f(0) = 2, and the absolute minimum is f(3) = -4.

Question 7

Find the derivative of y = (3x^2 + 2x)(4x - 1)^3.

Answer 7

The derivative is y’ = 3(4x - 1)^3 + (3x^2 + 2x)(12x - 3).

Question 8

Evaluate the integral of (2x + 1) / (x^2 + 3x + 2) with respect to x.

Answer 8

The integral evaluates to ln|2x + 1| - ln|x + 2| + C.

Question 9

Find the equation of the line tangent to the curve y = ln(4x) at the point where x = 1.

Answer 9

The equation of the tangent line is y = x - 1.

Question 10

Determine the average value of the function f(x) = x^2 on the interval [1, 3].

Answer 10

The average value is 7/3.

Question 11

Find the derivative of y = (x^2 - 1)(x^3 + 2x + 1).

Answer 11

The derivative is y’ = 5x^4 + 3x^2 + 2x - 1.

Question 12

Evaluate the integral of 1 / (x^2 - 4x + 4) with respect to x.

Answer 12

The integral evaluates to -1 / (x - 2) + C.

Question 13

Determine the value of c that satisfies the Mean Value Theorem for the function f(x) = x^3 - 3x^2 + 2x on the interval [0, 2].

Answer 13

The value of c is 4/3.

Question 14

Find the equation of the tangent plane to the surface z = x^2 + y^2 - 3x + 2y at the point (1, -1, 2).

Answer 14

The equation of the tangent plane is z = 2x + 2y + 1.

Question 15

Calculate the second derivative of f(x) = 4e^(2x) - 3x^2.

Answer 15

The second derivative is f’‘(x) = 16e^(2x) - 6.

Question 16

Find the derivative of y = (2x - 1)^3 e^x.

Answer 16

The derivative is y’ = (2x - 1)^3(e^x + 3e^x).

Question 17

Evaluate the integral of (2x + 1) / (x^2 - 3x + 2) with respect to x.

Answer 17

The integral evaluates to ln|x - 1| - ln|x - 2| + C.

Question 18

Determine the equation of the line normal to the curve y = 2x^3 + 3x - 1 at the point where x = -1.

Answer 18

The equation of the line is y = -11x + 3.

Question 19

Find the absolute maximum and minimum values of the function f(x) = 4x^3 - 3x^2 - 6x + 2 on the interval [-1, 2].

Answer 19

The absolute maximum is f(2) = 16, and the absolute minimum is f(-1) = -3.

Question 20

Evaluate the integral of cos^2(x) with respect to x.

Answer 20

The integral evaluates to (1/2)(x + sin(2x)) + C.

Question 21

Determine the critical points of the function f(x) = x^4 - 4x^3 + 6x^2 - 8x.

Answer 21

The critical points are x = 0, x = 2.

Question 22

Find the equation of the tangent line to the curve y = ln(x^2) at the point where x = e.

Answer 22

The equation of the tangent line is y = 2x - 1.

Question 23

Calculate the second derivative of g(x) = 3x^2 - 2x + 1.

Answer 23

The second derivative is g’‘(x) = 6.

Question 24

Evaluate the integral of (x^2 + 3x + 2) / (x^3 + x^2 + x) with respect to x.

Answer 24

The integral evaluates to (1/2) ln|x^2 + x| + 2 ln|x| - x + C.

Question 25

Find the derivative of y = (x^2 + 1)^(3/2) + (2x^3 + 3x)^(4/3)

Answer 25

The derivative is y’ = 3x(x^2 + 1)^(1/2) + 4x^2(2x^3 + 3x)^1/3.

Question 26

Consider the region bounded by the x-axis, the graph of y = 2x, and the line x = 2. Find the volume of the solid generated when this region is revolved around the x-axis.

Answer 26

The volume of the solid is 8π cubic units.

Question 27

Given the curve y = x^3 and the x-axis, determine the volume of the solid formed when the region bounded by this curve and the lines x = 0 and x = 1 is rotated around the y-axis.

Answer 27

The volume of the solid formed when the region is rotated around the y-axis is (π/4) cubic units.

Question 28

Find the volume of the solid obtained by revolving the region bounded by the curves y = x^2 and y = 2x around the line y = -1.

Answer 28

The volume of the solid is 56π/5 cubic units.

Question 29

The region bounded by the curves y = e^x, y = 0, x = 0, and x = 1 is revolved around the y-axis. Calculate the volume of the resulting solid.

Answer 29

The volume of the solid is 2π(1 - 1/e) cubic units.

Question 30

Consider the region bounded by the graph of y = x^2, the x-axis, and the lines x = 1 and x = 3. Find the volume of the solid formed when this region is revolved around the line y = -1.

Answer 30

The volume of the solid is 56π/5 cubic units.