Free Printable T-Level Eng – Maths - Differentiation Calculus Quiz

1.

In calculus for engineering, which statement best describes the derivative of a function y = f(x) with respect to x?

a)

It gives the average value of the function over an interval.

b)

It gives the instantaneous rate of change (gradient) of the function at a point.

c)

It gives the area under the graph of the function.

d)

It gives a fixed change in the function per unit time.

2.

Which of the following best describes a stationary point on the graph of y = f(x)?

a)

A point where the curve crosses the x-axis.

b)

A point where the gradient (derivative) is zero.

c)

A point where the derivative is infinite.

d)

A point where y is equal to zero.

3.

If a function has a positive derivative at x = a, what does this tell you about the function near x = a?

a)

The function is decreasing at x = a.

b)

The function is increasing at x = a.

c)

The function is stationary at x = a.

d)

The function is undefined at x = a.

4.

Which of the following is the correct general rule for differentiating y = $a x^n$ with respect to x? (a and n are constants.)

a)

$dy/dx = a n x^{(n-1)}$

b)

$dy/dx = a x^{(n-1)}$

c)

$dy/dx = n x^{(n-1)}$

d)

dy/dx = a n x^n

5.

Which of the following is a correct standard derivative?

a)

d/dx(cos(ax)) = a cos(ax)

b)

d/dx(sin(ax)) = a cos(ax)

c)

$d/dx(tan(x)) = sin^2(x)$

d)

$d/dx(e^{kx}) = e^{kx}$

6.

For the function y = $x^2$ , at what value of x does a stationary point occur?

a)

x = -1

b)

x = 0

c)

x = 1

d)

x = 2

7.

If s(t) represents the displacement of a moving object as a function of time t, what physical quantity does ds/dt represent?

a)

Acceleration of the object.

b)

Velocity of the object.

c)

Mass of the object.

d)

Average speed of the object over a time interval.

8.

If s(t) is displacement as a function of time, what physical quantity is represented by $d^2s/dt^2$ ?

a)

Force.

b)

Velocity.

c)

Acceleration.

d)

Power.

9.

Differentiate y = $5x^4$ with respect to x.

a)

$20x^3$

b)

$5x^3$

c)

$20x^4$

d)

$4x^5$

10.

Differentiate y = $3x^{-2}$ with respect to x.

a)

$-6x^{-3}$

b)

$6x^{-3}$

c)

$-6x^{-2}$

d)

$3x^{-3}$

11.

Differentiate y = $e^{3x}$ with respect to x.

a)

$e^{3x}$

b)

$3e^{3x}$

c)

$e^{x}$

d)

$3e^{x}$

12.

Differentiate y = $4e^{0.5x}$ with respect to x.

a)

$2e^{0.5x}$

b)

$4e^{0.5x}$

c)

$0.5e^{0.5x}$

d)

$2e^{x}$

13.

Differentiate y = $7\sin(2x)$ with respect to x.

a)

$14\cos(2x)$

b)

$7\cos(2x)$

c)

$-14\sin(2x)$

d)

$14\sin(2x)$

14.

Differentiate y = $5\cos(3x)$ with respect to x.

a)

$-15\sin(3x)$

b)

$15\sin(3x)$

c)

$-5\sin(3x)$

d)

$-15\cos(3x)$

15.

Differentiate y = $2\tan(x)$ with respect to x.

a)

$2\sec^2(x)$

b)

$2\sec(x)$

c)

$2\sin(x)$

d)

$2\cos(x)$

16.

Differentiate y = $x^3 + 4e^{x}$ with respect to x.

a)

$3x^2 + 4e^{x}$

b)

$3x^2 + e^{x}$

c)

$x^2 + 4e^{x}$

d)

$3x + 4e^{x}$

17.

A CNC milling tool moves in the x-direction so that its displacement (in mm) is given by $s(t) = 2t^3 - 5t$ , where t is the time in seconds. What is the instantaneous velocity of the tool at t = 2 s?

a)

11 mm/s

b)

19 mm/s

c)

29 mm/s

d)

35 mm/s

18.

An F1 car’s velocity in a straight-line test is given by $v(t) = 10t^2$ m/s, where t is the time in seconds after the start. What is the car’s acceleration at t = 3 s?

a)

30 m/s^2

b)

40 m/s^2

c)

50 m/s^2

d)

60 m/s^2

19.

A test rocket’s vertical position is modelled by $h(t) = 1000 + 50t - 4t^2$ , where h is in metres and t is in seconds. What is the vertical speed of the rocket at t = 5 s? (Upwards speed is positive.)

a)

-10 m/s

b)

0 m/s

c)

10 m/s

d)

90 m/s

20.

A rocket engine’s thrust is given by $T(t) = 200e^{0.1t}$ (in newtons). What is the rate of change of thrust with respect to time t?

a)

$20e^{0.1t}$

b)

$200e^{0.1t}$

c)

$0.1e^{0.1t}$

d)

$200e^{t}$

21.

An investment grows according to $A(t) = 5000e^{0.06t}$ , where A is the amount in pounds and t is the time in years. What is the instantaneous rate of growth of the investment?

a)

$0.06e^{0.06t}$

b)

$300e^{0.06t}$

c)

$5000e^{0.06t}$

d)

$300e^{t}$

22.

The angular position of a satellite in orbit is given by $\theta(t) = 0.5t^2$ (in radians), where t is time in seconds. What is the angular velocity at t = 4 s?

a)

0.5 rad/s

b)

2 rad/s

c)

4 rad/s

d)

8 rad/s

23.

A drone’s altitude is modelled by $h(t) = 3t^2 + e^{t}$ , where h is in metres and t is in seconds. Which expression gives the drone’s vertical acceleration?

a)

$6 + e^{t}$

b)

$6t + e^{t}$

c)

$6 + te^{t}$

d)

$e^{t} - 6$

24.

The displacement of a sewing machine head in a straight line is given by $s(t) = 4t^2 - 3t$ (in mm), where t is the time in seconds. What is the constant acceleration of the machine head?

a)

4 mm/s^2

b)

8 mm/s^2

c)

-3 mm/s^2

d)

8t mm/s^2

25.

The position of a robotic linear actuator is given by $x(t) = 6 - 2e^{-0.5t}$ (in mm), where t is in seconds. What is the actuator’s velocity as a function of time?

a)

$e^{-0.5t}$

b)

$-e^{-0.5t}$

c)

$2e^{-0.5t}$

d)

$-2e^{-0.5t}$

26.

The output voltage of an AC circuit is modelled by $V(t) = 10\sin(100t)$ , where V is in volts and t is in seconds. What is dV/dt?

a)

$100\cos(100t)$

b)

$1000\cos(100t)$

c)

$1000\sin(100t)$

d)

$-1000\sin(100t)$

27.

The flow rate of a pump is given by $Q(t) = 5 + 0.2t^3$ (in L/s). What is the rate of change of flow at t = 2 s?

a)

0.6 L/s^2

b)

1.2 L/s^2

c)

2.4 L/s^2

d)

3.0 L/s^2

28.

The pitch angle of a wind turbine blade is modelled by $\theta(t) = 15 - 4e^{-0.2t}$ degrees. What is d\theta/dt?

a)

$0.8e^{-0.2t}$

b)

$-0.8e^{-0.2t}$

c)

$4e^{-0.2t}$

d)

$-4e^{-0.2t}$

29.

The position of a train along a straight track is given by $x(t) = 100t - 2t^2$ (in metres). At what time t is the train momentarily stationary (velocity equal to zero)?

a)

10 s

b)

25 s

c)

50 s

d)

100 s

30.

The price of a company’s shares is modelled by $S(t) = 20 + 3t + 0.5t^2$ , where S is in pounds and t is time in months. At t = 4 months, what is the instantaneous rate of change of the share price?

a)

5 pounds/month

b)

6 pounds/month

c)

7 pounds/month

d)

8 pounds/month

T-Level Eng – Maths - Differentiation Calculus Quiz

Total questions: 30

Worksheet time: 15mins

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