Applications of Differentiation: Enhance Your Mathematical Modelling Skills

Question 1

The maximum value of the function f(x) = x³ - 3x + 2 on the closed interval [0, 2] is:

Accepted Answer

3

Answer

-1

Answer

1

Answer

5

Question 2

The rate of change of the area of a circle with respect to its radius r is:

Accepted Answer

2πr

Answer

πr²

Answer

4πr²

Answer

πr

Question 3

A rectangular box with a square base and an open top has a volume of 500 cm³. If x cm is the length of each side of the square base, the surface area of the box is minimized when x is equal to:

Accepted Answer

10

Answer

20

Answer

5

Answer

15

Question 4

Determine the equation of the tangent to the curve y = e^x at the point (0, 1).

Accepted Answer

y = x + 1

Answer

y = e^x

Answer

y = x + e

Answer

y = 1

Question 5

Find the maximum value of the function f(x) = -x² + 4x + 3.

Accepted Answer

5

Answer

-5

Answer

1

Answer

3

Question 6

Identify the function that describes the rate of change of the volume of a sphere with respect to its radius.

Accepted Answer

4πr²

Answer

πr

Answer

2πr

Answer

πr³

Question 7

Find the derivative of the function f(x) = √(x + 1).

Accepted Answer

1 / (2√(x + 1))

Answer

√(x + 1)

Answer

2√(x + 1)

Answer

1 / √(x + 1)

Question 8

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

Accepted Answer

1/5 cm/s

Answer

1/10 cm/s

Answer

5 cm/s

Answer

10 cm/s

Question 9

The function f(x) = x^3 - 3x^2 + 2x - 1 has a critical point at x = 1. What is the nature of this critical point?

Accepted Answer

Minimum

Answer

Point of inflection

Answer

Saddle point

Answer

Maximum

Question 10

What is the equation of the tangent line to the curve *y = x² + 1* at the point *(2, 5)*?

Accepted Answer

y = 4x - 3

Answer

y = 3x - 1

Answer

y = 2x + 1

Question 11

At what *x*-value does the function *f(x) = x³ - 3x² + 3x* have a point of inflection?

Accepted Answer

x = 1

Answer

x = 0

Answer

x = 2

Question 12

What is the *x*-coordinate of the local minimum of the function *f(x) = x³ - 6x² + 9x*?

Accepted Answer

x = 3

Answer

x = 2

Answer

x = -1

Answer

x = 1

Question 13

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

Accepted Answer

625 square meters

Answer

500 square meters

Answer

1000 square meters

Question 14

A projectile is launched vertically upwards with an initial velocity of 100 meters per second. What is the maximum height reached by the projectile?

Accepted Answer

1250 meters

Answer

750 meters

Answer

1000 meters

Answer

500 meters

Question 15

Find the critical points of the function f(x) = x^4 - 4x^3 + 3x^2 - 2x + 1.

Accepted Answer

x = 0, x = 1, x = 3

Answer

x = 1, x = 2, x = 3

Answer

x = 0, x = 1, x = 2

Answer

x = 0, x = 2, x = 4

Question 16

Find the inflection point(s) of the function f(x) = e^(-x^2).

Accepted Answer

x = 0

Answer

No inflection point exists

Answer

x = 1

Answer

x = -1

Question 17

A company estimates that the profit, in pounds, from selling x units of a product is given by P(x) = -x^3 + 12x^2 - 30x + 20. Find the number of units that should be sold to maximize profit.

Accepted Answer

8 units

Answer

16 units

Answer

4 units

Answer

12 units

Question 18

The velocity of a car is given by v(t) = 2t³ - 12t² + 18t, where t is time in seconds. What is the acceleration of the car at t = 2 seconds?

Accepted Answer

-12

Answer

12

Answer

-6

Answer

6