Calculus for Year 10 Secondary School Students in the UK: Interactive Quizzes and Learning Resources

Question 1

Consider the function f(x) = (x² + 1) / (x - 1). Determine if it's continuous at x = 1. If not, specify if the discontinuity is removable.

Accepted Answer

Removable discontinuity at x = 1

Answer

Continuous at x = 1

Answer

Infinite discontinuity at x = 1

Answer

Jump discontinuity at x = 1

Question 2

What is the definition of a limit in calculus?

Accepted Answer

The value that a function approaches as the input approaches a certain value

Answer

The value that a function takes at a specific input value

Answer

The slope of a function at a specific input value

Answer

The area under a curve between two points

Question 3

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

Accepted Answer

2x

Answer

x

Answer

x^2

Answer

2x^2

Question 4

What is the indefinite integral of the function f(x) = sin(x)?

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 5

Find the derivative of f(x) = x³ + 2x² + x + 1.

Accepted Answer

3x² + 4x + 1

Answer

x³ + 4x + 1

Answer

3x² + 2x + 1

Answer

x³ + 2x + 1

Question 6

Given the velocity function v(t) = t³ - 2t² + 3, find the particle's acceleration at time t = 1.

Accepted Answer

1 m/s²

Answer

-1 m/s²

Answer

0 m/s²

Answer

2 m/s²

Question 7

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

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x² - 10x + 1

Answer

3x² + 2x - 5

Answer

x³ + 2x² - 5

Question 8

Calculate the area under the curve of the function f(x) = x² from x = 0 to x = 2 using the definite integral.

Accepted Answer

8/3

Answer

4

Answer

16/3

Answer

8

Question 9

Determine the maximum and minimum values of the function f(x) = -x² + 4x + 3 on the interval [0, 3].

Accepted Answer

Maximum: 7, Minimum: 3

Answer

Maximum: 3, Minimum: 1

Answer

Maximum: 1, Minimum: -3

Answer

Maximum: 5, Minimum: -1

Question 10

Find the area under the curve of f(x) = x² between x = 0 and x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 11

Use the Mean Value Theorem to find a value c in the interval (0, 1) such that f'(c) = (f(1) - f(0)) / (1 - 0) for f(x) = x³ - 2x² + 1.

Accepted Answer

2/3

Answer

1/3

Answer

1/2

Answer

1

Question 12

Revised: The limit of a function f(x) as x approaches a is the value that f(x)

Accepted Answer

approaches as x gets closer and closer to a

Answer

equals when x equals a

Answer

is equal to the slope of the tangent line at x = a

Answer

is equal to the area under the curve from x = a to x = b

Question 13

Revised: Find the derivative of f(x) = (x+1)(x-2).

Accepted Answer

x^2 - x - 2

Answer

2x - 1

Answer

2

Answer

x - 1

Question 14

Revised: Which of the following is the antiderivative of f(x) = cos(x)?

Accepted Answer

sin(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Answer

-cos(x) + C

Question 15

Revised: Find the integral of the function f(x) = 1 / x

Accepted Answer

ln(x) + C

Answer

x + C

Answer

-1 / x + C

Answer

-ln(x) + C

Question 16

The limit of a function f(x) as x approaches a is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a.

Accepted Answer

The value that f(x) gets arbitrarily close to as x gets arbitrarily close to a

Answer

The value of f(a)

Answer

The value of f(x) when x = a

Answer

The slope of the graph of f(x) at x = a

Question 17

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

Accepted Answer

2x + 2

Answer

2x

Answer

x + 2

Answer

x

Question 18

The power rule for derivatives states that if f(x) = x^n, then f'(x) is:

Accepted Answer

nx^(n-1)

Answer

e^x

Answer

1/x

Answer

cos(x)

Question 19

The volume of a sphere is given by V = (4/3)πr^3. What is the rate of change of the volume with respect to the radius?

Accepted Answer

4πr^2

Answer

πr^3

Answer

2πr

Answer

πr^2

Question 20

The chain rule for derivatives states that if f(x) = g(h(x)), then f'(x) is:

Accepted Answer

g'(h(x)) * h'(x)

Answer

g'(x) * h'(x)

Answer

g'(h(x))

Answer

h'(x)

Question 21

Find the derivative of the function f(x) = e^(x^2).

Accepted Answer

2xe^(x^2)

Answer

e^(x^2)

Answer

x*e^(x^2-1)

Answer

(x^2)*e^(x^2)

Question 22

Find the limit of the function f(x) = (x² - 1) / (x - 1) as x approaches 1.

Accepted Answer

1

Answer

0

Answer

2

Answer

Does not exist

Question 23

Find the area under the curve y = x² + 1 between x = 0 and x = 2.

Accepted Answer

8/3 square units

Answer

4/3 square units

Answer

2/3 square units

Answer

1 square unit

Question 24

Which of the following functions has a negative second derivative?

Accepted Answer

y = x³

Answer

y = x²

Answer

y = x

Answer

y = e^x

Question 25

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

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² + 2x - 5

Answer

x³ + 4x - 5

Question 26

What is the definition of the integral of a function f(x)?

Accepted Answer

The area under the curve of f(x) from a to b

Answer

The rate of change of f(x) with respect to x

Answer

The slope of the graph of f(x) at x

Answer

The derivative of f(x) with respect to x

Question 27

Which of the following is NOT an application of integrals in the real world?

Accepted Answer

Predicting the future value of an investment

Answer

Calculating the volume of a solid

Answer

Determining the work done by a force

Answer

Calculating the center of mass of an object

Question 28

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

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

x + 3

Answer

2x

Question 29

Find the equation of the tangent line to the curve of f(x) = sin(x) at the point x = π/2.

Accepted Answer

y = 1

Answer

y = x - π/2

Answer

y = -x

Answer

y = cos(x)

Question 30

What is the antiderivative of f(x) = e^x?

Accepted Answer

e^x + C

Answer

x * e^x + C

Answer

x^2 * e^x + C

Answer

e^x / x + C

Question 31

State the chain rule for differentiation.

Accepted Answer

d/dx(f(g(x))) = f'(g(x)) * g'(x)

Answer

d/dx(f(g(x))) = g'(x) / f'(x)

Answer

d/dx(f(g(x))) = f'(x) + g'(x)

Answer

d/dx(f(g(x))) = f(g(x)) * g(x)

Question 32

Find the indefinite integral of the function f(x) = cos(x).

Accepted Answer

sin(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Answer

xcos(x) + C

Question 33

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

Accepted Answer

-e^(-x)

Answer

e^(-x)

Answer

-x * e^(-x)

Answer

e

Question 34

Which of the following applications of calculus is used to find the acceleration of an object in motion?

Accepted Answer

Differentiation

Answer

Integration

Answer

Substitution

Answer

Chain rule

Question 35

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

Accepted Answer

2 * cos(2x)

Answer

cos(2x)

Answer

2 * sin(2x)

Answer

sin(x)

Question 36

Which of the following is the correct chain rule formula?

Accepted Answer

d/dx(f(g(x))) = f'(g(x))⋅g'(x)

Answer

d/dx(f(g(x))) = g'(f(x))⋅f'(x)

Answer

d/dx(f(g(x))) = f'(x)⋅g(x)

Answer

d/dx(f(g(x))) = g(x)⋅f'(x)

Question 37

Which of the following integrals represents the area under the curve y = sin(x) from x = 0 to x = π?

Accepted Answer

∫[0,π] sin(x) dx

Answer

∫[π,0] sin(x) dx

Answer

-∫[0,π] sin(x) dx

Answer

-∫[π,0] sin(x) dx

Question 38

Which of the following is NOT a derivative rule?

Accepted Answer

Integration by parts

Answer

Product rule

Answer

Quotient rule

Answer

Chain rule

Question 39

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

Accepted Answer

e^x + C

Answer

-e^x + C

Answer

1/e^x + C

Answer

-1/e^x + C

Question 40

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

Accepted Answer

6x^2 - 10x + 3

Answer

4x^3 - 10x^2 + 3

Answer

2x^3 - 5x^2 + 3

Answer

6x^2 - 10x + 1

Question 41

Provide the integral formula for the function f(x) = cos(x).

Accepted Answer

sin(x) + C

Answer

cos(x) + C

Answer

tan(x) + C

Answer

cot(x) + C

Question 42

Which statement accurately describes a function with a negative second derivative?

Accepted Answer

Concave down

Answer

Concave up

Answer

Increasing

Answer

Decreasing

Question 43

Identify the order of the differential equation d^2y/dx^2 + y = x.

Accepted Answer

Second order

Answer

First order

Answer

Third order

Answer

Fourth order

Question 44

Which method can be used to solve the differential equation y' - 2y = x?

Accepted Answer

Both separation of variables and integrating factor

Answer

Separation of variables

Answer

Integrating factor

Answer

Variation of parameters

Question 45

Which of the following is NOT a limit law?

Accepted Answer

The chain rule

Answer

The limit of a sum is the sum of the limits

Answer

The limit of a product is the product of the limits

Answer

The limit of a difference is the difference of the limits

Question 46

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

Accepted Answer

3x^2 - 2

Answer

3x^2 + 2

Answer

x^3 - 2

Answer

x^2 - 1

Question 47

Evaluate the integral ∫(x^2 + 1) dx

Accepted Answer

x^3/3 + x + C

Answer

x^3 + x + C

Answer

2x^2 + x + C

Answer

x^2 + x + C

Question 48

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

Accepted Answer

e^x + C

Answer

x e^x + C

Answer

e^x/x + C

Answer

x e^x + 1/x + C

Question 49

Find the critical point(s) of the function f(x) = |x - 2|

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

The function has no critical points

Question 50

Which of the following is the indefinite integral of ∫(x^3 - 2x + 1) dx?

Accepted Answer

x^4/4 - x^2 + x + C

Answer

x^4 - 2x^2 + x + C

Answer

x^3/3 - x^2 + x + C

Answer

x^3 - 2x^2 + 1 + C

Question 51

Find the limit of (x² - 4) / (x - 2) as x approaches 0:

Accepted Answer

0

Answer

2

Answer

-2

Answer

4

Question 52

Which integral represents the area under the curve of f(x) = 2x + 1 from x = 0 to x = 2?

Accepted Answer

∫(2x + 1) dx from 0 to 2

Answer

∫(2x + 1) dx from 2 to 0

Answer

∫2x dx from 0 to 2 + ∫1 dx from 0 to 2

Answer

∫2x dx from 2 to 0 + ∫1 dx from 2 to 0

Question 53

Find the indefinite integral of f(x) = e^(2x):

Accepted Answer

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

Answer

e^(2x) + C

Answer

2e^(2x) + C

Answer

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

Question 54

What is the order of the differential equation y'' - 5y' + 6y = 0?

Accepted Answer

Second order

Answer

First order

Answer

Third order

Answer

Fourth order

Question 55

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

Accepted Answer

2x + 2

Answer

x^2 + 2

Answer

x + 2

Answer

1

Question 56

Calculate the area under the curve of the function y = x^2 between the points x = 0 and x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 57

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

Accepted Answer

y = -x + 1

Answer

y = x - 1

Answer

y = 2x

Answer

y = -2x

Question 58

Which of the following is the definition of a limit?

Accepted Answer

The value that a function approaches as its input value approaches a certain value

Answer

The rate of change of a function at a given point

Answer

The integral of a function from negative infinity to infinity

Answer

The area under the curve of a function from negative infinity to infinity

Question 59

Which of the following is the Chain Rule for derivatives?

Accepted Answer

f'(g(x)) = f(g(x)) * g'(x)

Answer

f'(g(x)) = g'(x) / f(x)

Answer

f'(g(x)) = f(x) / g'(x)

Answer

f'(g(x)) = g(x) * f'(x)

Question 60

Which of the following is the Fundamental Theorem of Calculus?

Accepted Answer

The integral of a derivative is the original function

Answer

The derivative of an integral is the original function

Answer

The limit of a sum is the sum of the limits

Answer

The product of two functions is the sum of their derivatives

Question 61

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

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 62

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

Accepted Answer

3x^2 + 4x - 5

Answer

5x^2 + 4x - 1

Answer

x^2 + 2x - 5

Question 63

Calculate the integral of the function f(x) = 2x + 3 with respect to x.

Accepted Answer

x^2 + 3x + C

Answer

2x + 3x + C

Answer

x^2 + C

Question 64

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

Accepted Answer

y = 2x - 1

Answer

y = x + 1

Answer

y = x^2 + 1