Master Derivatives Applications: Enhance Your Calculus Skills

Question 1

Which method is most appropriate for solving the differential equation dy/dx = xy ?

Accepted Answer

Separation of variables

Answer

Exact equations

Answer

Integration by parts

Answer

Substitution

Question 2

Find the derivative of f(x) = x³ + 2x² - 1 with respect to x

Accepted Answer

3x² + 4x

Answer

x³ - 2x

Answer

x² + 4x

Answer

3x - 2

Question 3

Determine the slope of the tangent to the curve y = sin(x) at x = π/4

Accepted Answer

√2 / 2

Answer

-1

Answer

1

Answer

0

Question 4

Determine the equation of the normal to the curve y = x² at the point (1, 1)

Accepted Answer

y = -x + 2

Answer

y = x + 2

Answer

y = x - 2

Answer

y = -x - 2

Question 5

Construct the differential equation that models the population growth of a bacteria colony where the growth rate is proportional to the population size

Accepted Answer

dy/dt = ky

Answer

d²y/dt² = k/y

Answer

dy/dt = k/y

Answer

d²y/dt² = ky

Question 6

Calculate the acceleration of an object moving along a straight line if its velocity is given by v = t³ - 2t

Accepted Answer

3t² - 2

Answer

3t + 2

Answer

t² - 2

Answer

t³ - 2

Question 7

A rectangular garden has a length that is twice its width. If the area of the garden is fixed at 100 m², determine the maximum possible perimeter

Accepted Answer

40 m

Answer

20 m

Answer

60 m

Question 8

A spherical balloon is being inflated at a constant rate of 10 cubic centimeters per second. Find the rate of change of the balloon's radius when its radius is 5 centimeters.

Accepted Answer

2/5 cm/s

Answer

-2/5 cm/s

Answer

5/2 cm/s

Answer

1/10 cm/s

Question 9

The marginal revenue function for a product is given by MR(x) = 100 - 0.5x, where x is the number of units sold. What is the total revenue from selling the first 100 units?

Accepted Answer

9500 pounds

Answer

11000 pounds

Answer

9000 pounds

Answer

10000 pounds

Question 10

A particle moves along a straight line with an acceleration of a(t) = t - 2 m/s^2. If the particle's initial velocity is 5 m/s, what is its velocity at t = 3 seconds?

Accepted Answer

8 m/s

Answer

10 m/s

Answer

6 m/s

Answer

12 m/s

Question 11

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

Accepted Answer

f'(x) = 3x² - 4x

Answer

f'(x) = 2x - 4

Answer

f'(x) = x² - 2

Answer

f'(x) = x³ - 4x

Question 12

Find the equation of the tangent line to the curve y = x² at the point (1, 1).

Accepted Answer

y = 2x - 1

Answer

y = x + 1

Answer

y = x² + 1

Answer

y = 2

Question 13

The derivative of the function f(x) = e^x is:

Accepted Answer

e^x

Answer

1/e^x

Answer

x*e^x

Answer

0

Question 14

Find the equation of the normal to the curve y = x² + 2x - 3 at the point (1, 0).

Accepted Answer

y = -x + 1

Answer

y = x + 1

Answer

y = -x - 1

Answer

y = x - 1

Question 15

Find the differential equation of the family of curves whose gradient at any point (x, y) is equal to 2x + y.

Accepted Answer

dy/dx = 2x + y

Answer

dy/dx = x + 2y

Answer

dy/dx = y - 2x

Answer

dy/dx = 2y - x

Question 16

A particle's position is given by s(t) = t³ - 6t² + 9t. At what time is the particle instantaneously at rest?

Accepted Answer

t = 1 or t = 3

Answer

t = 2

Answer

t = 0

Answer

t = 4

Question 17

The function f(x) = x³ - 3x² + 2 has a local maximum at x = ?

Accepted Answer

x = 0

Answer

x = 2

Answer

x = 1

Question 18

The rate of change of the volume of a sphere with respect to its radius is:

Accepted Answer

4πr²

Answer

πr²

Answer

2πr²

Answer

(4/3)πr³

Question 19

If the acceleration of a particle is given by a(t) = 2t, and its initial velocity is v(0) = 1, what is the velocity function v(t)?

Accepted Answer

v(t) = t² + 1

Answer

v(t) = 2t + 1

Answer

v(t) = 2t

Answer

v(t) = t²

Question 20

The critical points of the function f(x) = x⁴ - 4x³ + 4x² are located at:

Accepted Answer

x = 0 and x = 2

Answer

x = 0 and x = 1

Answer

x = 1 and x = 3

Answer

x = 1 and x = 2

Question 21

The equation of the tangent line to the curve y = ln(x) at the point (1, 0) is:

Accepted Answer

y = x - 1

Answer

y = -x + 1

Answer

y = x + 1

Question 22

The function f(x) = x³ - 3x² + 2 has a critical point at x = 1. Using the second derivative test, what is the nature of this critical point?

Accepted Answer

Local Maximum

Answer

Local Minimum

Answer

Inflection Point

Answer

Absolute Maximum

Question 23

The rate of change of a population P(t) is directly proportional to the size of the population itself.  Which differential equation accurately models this scenario?

Accepted Answer

dP/dt = kP

Answer

dP/dt = k

Answer

dP/dt = kP²

Answer

dP/dt = k/P

Question 24

What is the equation of the normal line to the curve y = ln(x) at the point (e, 1)?

Accepted Answer

y = -x + e + 1

Answer

y = x + e - 1

Answer

y = x - e + 1

Answer

y = -x - e - 1

Question 25

Which of the following is NOT a typical application of derivatives in real-world scenarios?

Accepted Answer

Determining the number of words in a book

Answer

Predicting the trajectory of a projectile

Answer

Optimizing the design of a container to minimize material use

Answer

Analyzing the rate of change of a population

Question 26

A particle moves along a straight line. Its displacement, *s* metres, from a fixed point O at time *t* seconds is given by *s = t³ - 6t² + 9t*.  For what values of *t* is the particle at rest?

Accepted Answer

t = 1 second and t = 3 seconds

Answer

t = 9 seconds only

Answer

t = 0 seconds only

Answer

t = 2 seconds only

Question 27

The function f(x) = 2x³ - 9x² + 12x - 3 has stationary points when *x* equals:

Accepted Answer

x = 1 and x = 2

Answer

x = -1 and x = -2

Answer

x = 3 only

Answer

x = 0 only

Question 28

A ball is thrown vertically upwards. Its height *h* metres after *t* seconds is given by *h = 20t - 5t²*. What is the maximum height reached by the ball?

Accepted Answer

20 metres

Answer

40 metres

Answer

15 metres

Answer

25 metres

Question 29

What is the equation of the tangent to the curve y = x² - 3x + 2 at the point (1, 0)?

Accepted Answer

y = -x + 1

Answer

y = x + 1

Answer

y = -x - 1

Answer

y = 2x - 2

Question 30

Which of the following is NOT an application of derivatives?

Accepted Answer

Finding the sum of an infinite geometric series

Answer

Determining the maximum or minimum values of a function

Answer

Finding the instantaneous rate of change of a function

Answer

Approximating the value of a function using linear approximation

Question 31

The function f(x) = x³ - 6x² + 9x has a point of inflection when *x* equals:

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 3

Answer

x = 1

Question 32

The velocity of a particle moving along a straight line is given by *v(t) = 3t² - 12t + 9*.  For what value of *t* is the particle's acceleration zero?

Accepted Answer

t = 2

Answer

t = 1

Answer

t = 1 and t = 3

Answer

t = 0

Question 33

A particle moves along a straight line with velocity v(t) = 2t - 3. What is the displacement of the particle from time t = 0 to t = 2?

Accepted Answer

2

Answer

-2

Answer

4

Answer

-4

Question 34

What are the co-ordinates of the critical point of the function f(x) = x³ - 3x² + 2?

Accepted Answer

(1, 0)

Answer

(2, -2)

Answer

(0, 2)

Answer

(-1, 0)

Question 35

What is the rate of change of the volume of a sphere with respect to its radius *r*?

Accepted Answer

4πr²

Answer

πr²

Answer

2πr

Answer

4πr

Question 36

A particle's position is given by s(t) = t³ - 6t² + 9t. What is the particle's velocity when t = 2?

Accepted Answer

0

Answer

-3

Answer

12

Answer

3

Question 37

The function f(x) = x³ - 3x² + 2 has a local maximum at what x-value?

Accepted Answer

0

Answer

1

Answer

3

Answer

2

Question 38

A rectangular garden is to be enclosed by 100 meters of fencing. What is the maximum possible area of the garden?

Accepted Answer

625 square meters

Answer

500 square meters

Answer

1000 square meters

Answer

2500 square meters

Question 39

What is the equation of the tangent line to the curve y = x² + 1 at the point (1, 2)?

Accepted Answer

y = 2x

Answer

y = 2x + 1

Answer

y = x + 1

Answer

y = x + 2

Question 40

The rate of change of a population is proportional to the population itself. If the initial population is 100 and it doubles in 5 years, what is the population after 10 years?

Accepted Answer

400

Answer

1600

Answer

800

Answer

200

Question 41

A ball is thrown vertically upwards with an initial velocity of 20 m/s. What is the ball's maximum height?

Accepted Answer

20.41 meters

Answer

10.20 meters

Answer

61.23 meters

Answer

40.82 meters

Question 42

What is the critical point of the function f(x) = x^4 - 4x^3?

Accepted Answer

(0, 0) and (3, -27)

Answer

(1, -3) and (3, -27)

Answer

(2, -16) and (3, -27)

Answer

(0, 0) and (2, -16)

Question 43

What is the concavity of the function f(x) = x^3 - 3x^2 + 2 at the point (1, 0)?

Accepted Answer

Concave Down

Answer

Concave Up

Answer

Point of Inflection

Answer

None of the above

Question 44

A company's profit function is given by P(x) = -x² + 100x - 1000, where x is the number of units produced. What is the maximum profit?

Accepted Answer

2500

Answer

500

Answer

1000

Answer

1500

Question 45

The function f(x) = 2x³ - 6x² + 4x has an inflection point at what x-value?

Accepted Answer

1

Answer

2

Answer

0

Answer

3

Question 46

A particle moves along a straight line with its position given by s(t) = t³ - 6t² + 9t. At what time(s) does the particle come to rest?

Accepted Answer

t = 1 and t = 3

Answer

t = 1 only

Answer

t = 0 and t = 3

Answer

t = 3 only

Question 47

The function f(x) = x³ - 3x² + 2 has a local maximum at x = ?

Accepted Answer

x = 0

Answer

x = 1

Answer

x = -1

Answer

x = 2

Question 48

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

Accepted Answer

625 square meters

Answer

1000 square meters

Answer

2500 square meters

Answer

500 square meters

Question 49

What is the slope of the tangent line to the curve y = 2x² - 3x + 1 at the point (2, 3)?

Accepted Answer

5

Answer

7

Answer

2

Answer

3

Question 50

A company's profit function is given by P(x) = -x³ + 12x² - 36x, where x represents the number of units produced. How many units should the company produce to maximize its profit?

Accepted Answer

x = 4

Answer

x = 2

Answer

x = 6

Answer

x = 8

Question 51

The function f(x) = x⁴ - 4x³ has an inflection point at x = ?

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 3

Answer

x = 1

Question 52

The rate of change of a population is proportional to the population itself. If the population doubles in 5 years, what is the growth constant (k)?

Accepted Answer

k = (ln 2) / 5

Answer

k = 5 ln 2

Answer

k = ln 2

Answer

k = 2/5

Question 53

What is the equation of the line tangent to the curve y = e^x at the point (0, 1)?

Accepted Answer

y = x + 1

Answer

y = -x + 1

Answer

y = x - 1

Answer

y = 2x + 1

Question 54

A rocket is launched vertically with its height above ground given by h(t) = -16t² + 64t. What is the rocket's maximum height?

Accepted Answer

64 feet

Answer

32 feet

Answer

16 feet

Answer

128 feet

Question 55

The function f(x) = x² - 4x + 5 is decreasing on the interval ?

Accepted Answer

(2, ∞)

Answer

(0, 2)

Answer

(-∞, ∞)

Answer

(-∞, 2)