Calculus for Year 11: Master Differential and Integral Calculus Concepts

Question 1

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

Accepted Answer

$6x^2 - 5x - 3$

Answer

$4x^2 - 5x - 3$

Answer

$6x^2 + 5x - 3$

Answer

$4x^2 + 5x - 3$

Question 2

Determine the limit of the function $$rac{x^4 + 2x^2 - 1}{x^3 - 4x^2 + 2x}$$ as x approaches infinity.

Accepted Answer

Infinity

Answer

Zero

Answer

One

Question 3

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

Accepted Answer

0.4 cm/s

Answer

1 cm/s

Answer

2 cm/s

Question 4

Find the limit of the function as x approaches infinity: $$lim_{x 	o infty} left( 1 + rac{1}{x} ight)^x $$

Accepted Answer

e

Answer

0

Answer

1

Answer

Infinity

Question 5

Which of the following is the mathematical definition of the derivative of a function f(x)?

Accepted Answer

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

Answer

The gradient of the normal to the curve of f(x) at x.

Answer

The area under the curve of f(x) between two points.

Answer

The x-coordinate of the vertex of the graph of f(x).

Question 6

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

Accepted Answer

3x^2 - 10x + 2

Answer

3x^2 - 5x + 2

Answer

x^2 - 10x + 2

Answer

x^2 - 5x + 2

Question 7

Which of the following is NOT a property of the derivative?

Accepted Answer

The derivative of a quotient of functions is the sum of the derivatives.

Answer

The derivative of a constant function is zero.

Answer

The derivative of a sum of functions is the sum of the derivatives.

Answer

The derivative of a product of functions is the product of the derivatives.

Question 8

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

x² + 2x - 5

Question 9

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

Accepted Answer

2

Answer

0

Answer

4

Answer

Undefined

Question 10

Find the derivative of the function f(x) = |x|.

Accepted Answer

1 if x > 0, 0 if x = 0, -1 if x < 0

Answer

1 if x > 0, -1 if x < 0

Answer

x if x > 0, 0 if x = 0, -x if x < 0

Answer

Undefined at x = 0

Question 11

Which of the following limits does not exist?

Accepted Answer

lim_(x->0) (1/x)

Answer

lim_(x->0) (x^2)/x

Answer

lim_(x->∞) sin(x)

Answer

lim_(x->1) (x-1)/(x^2-1)

Question 12

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

Accepted Answer

3x^2 + 2

Answer

3x + 2

Answer

x^3 + 2

Answer

x^2 + 2x - 1

Question 13

Which of the following is an example of a function that has an absolute maximum but no absolute minimum?

Accepted Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = x

Question 14

What is the antiderivative of f(x) = cos(x)?

Accepted Answer

sin(x) + C

Answer

sin(x)

Answer

cos(x) + C

Answer

-sin(x)

Question 15

Find the derivative of f(x) = 2^x:

Accepted Answer

2^x * ln(2)

Answer

2^(x+1)

Answer

x * 2^(x-1)

Answer

ln(2) * 2^x

Question 16

Evaluate the integral of f(x) = sin(2x) from x = 0 to x = π/2:

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 17

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 + 2x - 5

Answer

x^3 + 4x^2 - 5x + 1

Question 18

Calculate the indefinite integral of the trigonometric function f(x) = sin(x).

Accepted Answer

cos(x) + C

Answer

-cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 19

Calculate the indefinite integral of the exponential function f(x) = e^x.

Accepted Answer

e^x + C

Answer

-e^x + C

Answer

x * e^x + C

Answer

-x * e^x + C

Question 20

In Calculus, which of the following is NOT a fundamental concept?

Accepted Answer

Exponents

Answer

Limits

Answer

Derivatives

Answer

Integrals

Question 21

Given f(x) = x^2 + 3, what is its derivative?

Accepted Answer

2x + 3

Answer

x^2

Answer

3

Answer

2x

Question 22

When using integration, which formula calculates the area under a curve?

Accepted Answer

∫ f(x) dx

Answer

∫ dx f(x)

Answer

ƒ(x) ∫ dx

Answer

dx ∫ f(x)

Question 23

As x approaches 1, what is the limit of (x^2 - 1) / (x - 1)?

Accepted Answer

1

Answer

0

Answer

2

Answer

Undefined

Question 24

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

Accepted Answer

-cos(x)

Answer

-sin(x)

Answer

cos(x)

Answer

sin(x)

Question 25

What is the correct formula for the derivative of f(x) = ln(x)?

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

1/ln(x)

Question 26

Determine the derivative of f(x) = sin(2x) with respect to x.

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

2sin(2x)

Answer

sin(2x)

Question 27

Find the limit of (e^x - 1) / x as x approaches infinity.

Accepted Answer

1

Answer

0

Answer

e

Answer

undefined

Question 28

Calculate the area of the region bounded by the curves y = x and y = x^2.

Accepted Answer

1/3

Answer

1/2

Answer

1

Answer

2

Question 29

Find the limit of (1 + 1/x)^x as x approaches infinity.

Accepted Answer

e

Answer

0

Answer

1

Answer

undefined

Question 30

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x + 4

Answer

x + 2

Question 31

Calculate the integral of f(x) = 2x + 1 from x = 0 to x = 2.

Accepted Answer

6

Answer

3

Answer

9

Answer

12

Question 32

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

Accepted Answer

x = -1, x = 1

Answer

x = 1, x = 2

Answer

x = -2, x = 0

Answer

x = 0, x = 2

Question 33

Identify the graph of the function f(x) = |x - 1|.

Accepted Answer

A V-shaped graph with vertex (1, 0)

Answer

A straight line with slope 1

Answer

A parabola with vertex (1, 0)

Answer

A circle with radius 1

Question 34

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

Accepted Answer

16π/3

Answer

8π/3

Answer

32π/3

Answer

64π/3

Question 35

Which of the following is the correct chain rule for derivatives?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 36

Find the indefinite integral of the function f(x) = e^x with respect to x.

Accepted Answer

e^x + C

Answer

x * e^x + C

Answer

x * e^(x-1) + C

Answer

e^(x-1) + C

Question 37

Calculate the integral of the function f(x) = cos(x).

Accepted Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Answer

-cos(x) + C

Question 38

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

Accepted Answer

2

Answer

0

Answer

4

Answer

6

Question 39

Identify which of the following is a second-order differential equation.

Accepted Answer

y'' + 3y' + 2y = 0

Answer

y'y'' + 2y' = 0

Answer

y''' + y' = 0

Answer

y^2 + y' = 0

Question 40

Identify which of the following is a vector function.

Accepted Answer

r(t) = (t^2, t^3)

Answer

f(x) = x^2 + 3x

Answer

g(x, y) = x^2 + y^2

Answer

h(x, y, z) = xyz

Question 41

In mathematics, what does the term 'limit' refer to?

Accepted Answer

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

Answer

The exact value of a function at a specific input

Answer

The rate of change of a function

Answer

The total area under the curve of a function

Question 42

For the function f(x) = x^2, what is its derivative?

Accepted Answer

2x

Answer

x^2

Answer

x

Answer

1

Question 43

Determine the integral of the function f(x) = 1/(x^2 + 1).

Accepted Answer

arctan(x) + C

Answer

tan^-1(x) + C

Answer

cot^-1(x) + C

Answer

arccot(x) + C

Question 44

Which rule describes the chain rule for derivatives?

Accepted Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).

Answer

If f(x) = g(h(x)), then f'(x) = h'(g(x)) * g'(x).

Answer

If f(x) = g(h(x)), then f'(x) = g(h'(x)) * h(x).

Answer

If f(x) = g(h(x)), then f'(x) = h(g'(x)) * g(x).

Question 45

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

Accepted Answer

cos(x^2) * 2x

Answer

cos(x^2)

Answer

sin(x^2) * 2x

Answer

sin(x^2)

Question 46

Which theorem is known as the Fundamental Theorem of Calculus, Part I?

Accepted Answer

If f(x) is continuous on the interval [a, b], then ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is any antiderivative of f(x).

Answer

If f(x) is differentiable on the interval [a, b], then ∫[a, b] f(x) dx = f(b) - f(a).

Answer

If f(x) is continuous on the interval [a, b], then ∫[a, b] f(x) dx = 1 / 2 * [f(a) + f(b)].

Answer

If f(x) is differentiable on the interval [a, b], then ∫[a, b] f(x) dx = 0.

Question 47

Which of the following graphs represents the function f(x) = |x|?

Accepted Answer

V-shaped graph

Answer

Exponential graph

Answer

Parabolic graph

Answer

Sine wave graph

Question 48

Calculate the area of the region bounded by the curves y = x^2 and y = 2x + 1 using integral calculus.

Accepted Answer

3/2

Answer

1/2

Answer

2

Question 49

Find the derivative of the inverse trigonometric function y = sin^-1(x) and explain its significance.

Accepted Answer

1 / √(1 - x^2)

Answer

1 / x

Answer

√(1 - x^2)

Answer

x / √(1 - x^2)

Question 50

State the Taylor series expansion of the function f(x) = e^x centered at x = 0 and explain its applications.

Accepted Answer

1 + x + x^2/2! + x^3/3! + ...

Answer

x + x^2/2! + x^4/4! + ...

Answer

1 + x^2/2! + x^3/3! + ...

Answer

1 + x + x^3/3! + x^4/4! + ...

Question 51

Find the indefinite integral of the function: f(x) = 2x + 3

Accepted Answer

x^2 + 3x + C

Answer

2x^2 + 3x

Answer

2x - 3x + C

Answer

x^2 - 3x + C

Question 52

Find the derivative of the function: f(x) = ln(x^2 + 1)

Accepted Answer

2x / (x^2 + 1)

Answer

1 / (x^2 + 1)

Answer

(x^2 + 1) / 2x

Answer

2x

Question 53

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

Accepted Answer

3x² + 4x

Answer

2x³ + x

Answer

x³ - 2x² + 1

Answer

x² + 4x + 3

Question 54

Determine the equation of the tangent line to the curve y = x² + 1 at the point (1, 2).

Accepted Answer

y = 3x - 1

Answer

y = x + 2

Answer

y = 2x

Answer

y = x² + 1

Question 55

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

Accepted Answer

1/x

Answer

ln(x)

Answer

x

Question 56

Find the equation of the normal line to the curve y = x³ + 1 at the point (1, 2).

Accepted Answer

y = -x + 3

Answer

y = x + 1

Answer

y = -x + 1

Answer

y = x + 2

Question 57

Evaluate f(g(x)) given f(x) = 2x + 5 and g(x) = x^2.

Accepted Answer

2x^2 + 5

Answer

4x + 5

Answer

2x + x^2

Answer

x^2 + 5

Question 58

Calculate the area under the curve y = sin(x) from x = 0 to x = π/2.

Accepted Answer

1

Answer

0

Answer

π/2

Answer

2

Question 59

Find the antiderivative of the function 1/(x^2 + 4).

Accepted Answer

(1/2)arctan(x/2) + C

Answer

ln(x^2 + 4) + C

Answer

1/x + C

Answer

x^2 + C

Question 60

Which of the following is the equation of the curve that satisfies the differential equation dy/dx = x^2 + 2x?

Accepted Answer

y = (x^3)/3 + x^2 + C

Answer

y = x^2 + 2x + C

Answer

y = x^3 + x + C

Answer

y = (x^3)/3 + C

Question 61

Find the inflection point of the function y = x^3 - 6x^2 + 9x.

Accepted Answer

(2, 1)

Answer

No inflection point

Answer

(0, 0)

Answer

(3, 0)

Question 62

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

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x² - 5x - 1

Answer

3x² - 2x - 5

Answer

x² + 4x - 5

Question 63

Select the graph that best represents the function f(x) = 1 / x.

Accepted Answer

Image of a hyperbola

Answer

Image of a circle

Answer

Image of a parabola

Answer

Image of a straight line

Question 64

The function f(x) = |x| has which characteristic at x = 0?

Accepted Answer

Continuous but not differentiable

Answer

Not continuous

Answer

Differentiable everywhere

Answer

Not defined