Calculus: Limits, Derivatives, Integrals, and Applications

Question 1

Determine the derivative of f(x) = sin(x^2) + e^x.

Accepted Answer

2x cos(x^2) + e^x

Answer

2x sin(x^2) + e^x

Answer

cos(x^2) + e^x

Answer

sin(x^2) + 2e^x

Question 2

According to Rolle's Theorem, if f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), which of the following must be true?

Accepted Answer

f'(c) = 0

Answer

f'(c) = (f(b) - f(a)) / (b-a)

Answer

f'(c) = f(a) + f(b)

Question 3

Use the squeeze theorem to find the limit of the expression:

lim_(x->0) (x^2) * sin(1 / x)

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not exist

Question 4

Calculate the derivative of the function: f(x) = sin(x^2 + 1)

Accepted Answer

cos(x^2 + 1) * 2x

Answer

sin(x^2 + 1)

Answer

2x * cos(x^2 + 1)

Answer

x * cos(x^2 + 1)

Question 5

Find the equation of the tangent line to the graph of f(x) = sin(x) at x = π/4.

Accepted Answer

y = √2/2x + √2/2π/4

Answer

y = x - π/4

Answer

y = √2/2x - √2/2π/4

Answer

y = x + π/4

Question 6

Use the product rule to find the derivative of the function f(x) = (x^2 + 3)(x - 1).

Accepted Answer

x^3 + 2x^2 - 3x - 3

Answer

2x^2 - 2x - 3

Answer

x^3 - x^2 + 3x - 3

Answer

x^2 + 2x - 3

Question 7

Find the velocity of a particle given its acceleration function a(t) = 6t - 2 and initial velocity v(0) = 3.

Accepted Answer

3t^2 - 2t + 3

Answer

6t^2 - 2t - 3

Answer

3t^2 + 2t + 3

Answer

6t^2 + 2t - 3

Question 8

What is the indefinite integral of the function f(x) = e^(2x)?

Accepted Answer

(1/2)e^(2x) + C

Answer

(1/2)e^(2x) - C

Answer

e^(2x) + C

Answer

e^(2x) - C

Question 9

Find the area under the curve of the function f(x) = x^2 from x = 0 to x = 2.

Accepted Answer

8/3

Answer

4/3

Answer

16/3

Answer

32/3

Question 10

Find the absolute maximum value of the function f(x) = x^2 - 4x + 3 on the interval [0, 3].

Accepted Answer

5

Answer

-5

Answer

-1

Answer

0

Question 11

Determine the derivative of the function f(x) = x³, where x is a real number.

Accepted Answer

3x²

Answer

x²

Answer

3x

Answer

1/x²

Question 12

Determine the indefinite integral of the function f(x) = 1/x, where x is a real number and x ≠ 0.

Accepted Answer

ln(|x|)

Answer

-ln(|x|)

Answer

x

Answer

1/x²

Question 13

Calculate the second derivative of the function f(x) = sin(x), where x is a real number.

Accepted Answer

-cos(x)

Answer

-sin(x)

Answer

cos(x)

Answer

sin(2x)

Question 14

Which of the following is the integral of the function f(x) = e^x?

Accepted Answer

e^x + C

Answer

sin(e^x) + C

Answer

x * e^x + C

Answer

e^(x^2) + C

Question 15

Solve the following optimization problem: Find the maximum value of the function f(x) = -x^2 + 4x + 3.

Accepted Answer

x = 2

Answer

x = 3

Answer

x = -2

Answer

x = 1

Question 16

Which of the following is the second derivative of the function f(x) = ln(x)?

Accepted Answer

1/x^2

Answer

1/x

Answer

-1/x

Question 17

Which of the following is an application of integrals in real-world scenarios?

Accepted Answer

Calculating the work done by a force over a distance

Answer

Finding the circumference of a circle

Answer

Determining the derivative of a function

Question 18

Find the equation of the tangent line to the curve y = x^3 - 2x^2 + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = -x + 1

Answer

y = x + 1

Answer

y = 2x - 1

Question 19

Solve the related rates problem: A ladder 10 feet long is leaning against a wall. The bottom of the ladder is sliding away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall?

Accepted Answer

8/3 feet per second

Answer

3/8 feet per second

Answer

6 feet per second

Answer

2 feet per second

Question 20

Which of the following is the equation of the normal line to the graph of the function f(x) = e^x at x = 0?

Accepted Answer

y = -x + 1

Answer

y = e^x - 1

Answer

y = x

Answer

y = e^x

Question 21

What is the derivative of the function *f(x) = x³ + 2x*?

Accepted Answer

3x² + 2

Answer

x + 2

Answer

3x + 2

Answer

x² + 2

Question 22

What is the area under the curve of the function *f(x) = x²* from *x = 0* to *x = 2*?

Accepted Answer

8/3

Answer

2/3

Answer

4

Answer

16

Question 23

Find the critical points of the function *f(x) = x⁴ - 2x²*.

Accepted Answer

x = 0, x = ±√2

Answer

x = ±1

Answer

x = √2

Answer

x = 0

Question 24

Which of the following is the differential equation for the function *y = e^x + C*?

Accepted Answer

y' = y

Answer

y' = y - 1

Answer

y' = y + 1

Answer

y' = -y

Question 25

Find the derivative of the function *f(x) = x³ - 2x² + 5x - 1*.

Accepted Answer

3x² - 4x + 5

Answer

3x - 4

Answer

x³ - 4x² + 10x - 5

Answer

x² - 4x + 5

Question 26

What is the derivative of the function *f(x) = sin(x)*?

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

cos(2x)

Answer

sin(x)

Question 27

Find the derivative of the function *f(x) = e^x*.

Accepted Answer

e^x

Answer

e^(x^2)

Answer

1/e^x

Answer

x*e^x

Question 28

What is the indefinite integral of the function *f(x) = 2x + 1*?

Accepted Answer

x² + x + C

Answer

1/2x² + C

Answer

x³ + C

Answer

2x² + C

Question 29

What is the definite integral of the function  *f(x) = sin(x)* from 0 to π/2?

Accepted Answer

1

Answer

0

Answer

2

Answer

π/2

Question 30

Which formula represents the volume of a solid formed by rotating the area under the curve *y = f(x)* from *x = a* to *x = b* around the x-axis?

Accepted Answer

π∫[a,b] f(x)^2 dx

Answer

π∫[a,b] f(x) dx

Answer

2π∫[a,b] f(x) dx

Answer

2π∫[a,b] f(x)^2 dx

Question 31

What is the derivative of the function *f(x) = sin(x^2)*?

Accepted Answer

2x cos(x^2)

Answer

cos(x^2)

Answer

-cos(2x)

Answer

2x sin(x^2)

Question 32

What is the limit of the function *f(x) = (x^2 - 4) / (x - 2)* as *x* approaches 2?

Accepted Answer

4

Answer

Does not exist

Answer

0

Answer

2

Question 33

Find the derivative of the function *f(x) = 3x^2 + 2x - 1*.

Accepted Answer

6x + 2

Answer

9x + 2

Answer

6x^2 + 2x

Answer

3x + 2

Question 34

A rectangular garden is to be enclosed by 100 feet of fencing. What is the maximum area that can be enclosed?

Accepted Answer

625 square feet

Answer

500 square feet

Answer

1000 square feet

Answer

2500 square feet

Question 35

A spherical balloon is being inflated at a rate of 10 cubic inches per second. How fast is the radius increasing when the radius is 5 inches?

Accepted Answer

1 / (5π) inches per second

Answer

10 / (5π) inches per second

Answer

2 / (5π) inches per second

Answer

5 / π inches per second

Question 36

Find the indefinite integral of *f(x) = 2x + 1*.

Accepted Answer

x^2 + x + C

Answer

2x^2 + x + C

Answer

2x + C

Answer

x^2 + 1 + C

Question 37

Calculate the definite integral of *f(x) = x^2* from *x = 0* to *x = 2*.

Accepted Answer

8/3

Answer

4/3

Answer

4

Answer

2

Question 38

Which definite integral represents the area under the curve of *f(x) = x* from *x = 0* to *x = 2*?

Accepted Answer

∫(0 to 2) x dx

Answer

∫(0 to 2) x^2 dx

Answer

∫(0 to 2) 1/x dx

Answer

∫(0 to 2) 2x dx

Question 39

The function f(x) = x⁴ - 4x³ + 6x² - 4x + 1 has a relative minimum at which x-coordinate?

Accepted Answer

1

Answer

3

Answer

0

Answer

2

Question 40

What is the derivative of f(x) = 3x² + 2x - 1?

Accepted Answer

6x + 2

Answer

9x + 2

Answer

6x² + 2x

Answer

3x + 2

Question 41

A ball is thrown vertically upward with an initial velocity of 30 meters per second. Its height (in meters) after t seconds is given by h(t) = 30t - 5t². What is the maximum height reached by the ball?

Accepted Answer

45 meters

Answer

30 meters

Answer

15 meters

Answer

60 meters

Question 42

What is the derivative of y = sin(x) with respect to x?

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Answer

-cos(x)

Question 43

The definite integral of f(x) = x from x = 0 to x = 2 represents:

Accepted Answer

The area under the curve of f(x) from x = 0 to x = 2

Answer

The slope of the line tangent to f(x) at x = 2

Answer

The total change in f(x) from x = 0 to x = 2

Answer

The average value of f(x) from x = 0 to x = 2

Question 44

Find the limit of the function f(x) = (x² - 4) / (x - 2) as x approaches 2.

Accepted Answer

4

Answer

undefined

Answer

2

Answer

0

Question 45

Find the derivative of the function f(x) = x³ - 2x² + 5x.

Accepted Answer

3x² - 4x + 5

Answer

3x² + 4x + 5

Answer

x³ - 2x + 5

Answer

x² - 2x + 5

Question 46

Find the critical points of the function f(x) = x² - 6x + 8.

Accepted Answer

x = 3

Answer

x = 2

Answer

x = -3

Answer

x = 8

Question 47

Find the area under the curve of the function f(x) = 2x + 1 from x = 0 to x = 2.

Accepted Answer

6

Answer

8

Answer

4

Answer

12

Question 48

Find the maximum value of the function f(x) = -x² + 4x + 3 on the interval [0, 3].

Accepted Answer

7

Answer

3

Answer

9

Answer

0

Question 49

If y = x^3 + 2x^2 - 1, what is the related rate of change of y with respect to x when x is changing at a rate of 2 units per second?

Accepted Answer

12x + 4

Answer

3x^2 + 4x

Answer

6x + 2

Answer

x^3 + 2x

Question 50

On the curve y = x^2 + 3x - 5, which point is closest to (1, 0)?

Accepted Answer

(-1, -3)

Answer

(2, -1)

Answer

(1, -1)

Question 51

Calculate the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 2x + 4 about the x-axis.

Accepted Answer

(80π/3) cubic units

Answer

(40π/3) cubic units

Answer

(32π/3) cubic units

Answer

(16π/3) cubic units

Question 52

A force F(x) = x^2 + 2x Newtons acts over a distance of 5 meters. Calculate the work done by this force.

Accepted Answer

(175/2) Joules

Answer

50 Joules

Answer

100 Joules

Answer

25 Joules

Question 53

If an object moves along a straight line with its position given by s(t) = t^3 - 6t^2 + 9t + 2, determine its acceleration at time t = 2 seconds.

Accepted Answer

2 m/s^2

Answer

4 m/s^2

Answer

8 m/s^2

Answer

6 m/s^2

Question 54

Evaluate the limit of the function f(x) = (x^2 - 1) / (x - 1) as x approaches 1.

Accepted Answer

2

Answer

0

Answer

Does not exist

Answer

1

Question 55

A particle moves along a straight line with its position given by s(t) = 2t^3 - 9t^2 + 12t - 4, where t is in seconds and s is in meters. Find the particle's velocity at t = 2 seconds.

Accepted Answer

0 m/s

Answer

12 m/s

Answer

-3 m/s

Answer

2 m/s

Question 56

Determine the derivative of the function f(x) = sin(x^2).

Accepted Answer

2x cos(x^2)

Answer

cos(x^2)

Answer

2x sin(x^2)

Answer

-2x cos(x^2)

Question 57

Calculate the area under the curve y = x^2 + 1 from x = 0 to x = 2.

Accepted Answer

14/3 square units

Answer

8/3 square units

Answer

4 square units

Answer

10/3 square units

Question 58

The radius of a circular oil spill increases at a rate of 2 meters per minute. Determine the rate at which the area of the spill increases when the radius is 10 meters.

Accepted Answer

40π square meters per minute

Answer

2π square meters per minute

Answer

10π square meters per minute

Answer

20π square meters per minute

Question 59

Find the critical points of the function f(x) = x^3 - 3x^2.

Accepted Answer

x = 0 and x = 2

Answer

x = 2 only

Answer

x = -2 and x = 2

Answer

x = 0 only

Question 60

What is the integral of the function f(x) = 2x + 5?

Accepted Answer

x^2 + 5x + C

Answer

2x^2 + 5x + C

Answer

x^2 + C

Question 61

A rectangular fence is to be built with one side along a river (requiring no fencing). If 100 meters of fencing are available, what dimensions maximize the enclosed area?

Accepted Answer

25 meters by 50 meters

Answer

50 meters by 50 meters

Answer

10 meters by 80 meters

Answer

20 meters by 60 meters

Question 62

What is the average value of the function f(x) = x^2 on the interval [0, 2]?

Accepted Answer

4/3

Answer

8/3

Answer

2

Answer

4

Question 63

Evaluate the limit: lim (x->∞) (3x^2 + 2x - 1) / (5x^2 - x + 4)

Accepted Answer

3/5

Answer

∞

Answer

0

Answer

-1/4

Question 64

Find the limit of (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

4

Answer

0

Answer

2

Answer

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