Calculus for Year 13: Master Differential and Integral Calculus

Question 1

Calculate the integral of the function f(x) = 1/x with respect to x.

Accepted Answer

ln(|x|) + C

Answer

-ln(|x|) + C

Answer

1/x + C

Answer

-1/x + C

Question 2

Which of the following limit definitions is correct for the derivative?

Accepted Answer

lim(h->0) [f(x + h) - f(x)]/h

Answer

lim(h->∞) [f(x + h) - f(x)]/h

Answer

lim(x->0) [f(x) - f(0)]/x

Answer

lim(x->∞) [f(x) - f(0)]/x

Question 3

Evaluate the integral ∫(e^x + 1)/(e^x - 1) dx.

Accepted Answer

e^x + ln|e^x - 1| + C

Answer

e^x - ln|e^x + 1| + C

Answer

-e^x + ln|e^x - 1| + C

Answer

-e^x - ln|e^x + 1| + C

Question 4

State the chain rule for differentiation.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 5

Evaluate the integral ∫(x^2 + 3x - 2) dx.

Accepted Answer

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

Answer

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

Answer

2x^3 + 9x^2 - 6x + C

Answer

2x^3 + 3x^2 - 6x + C

Question 6

Use the Chain Rule to find the derivative of the function f(x) = (x^2 + 1)^3.

Accepted Answer

3(x^2 + 1)^2 * 2x

Answer

6x(x^2 + 1)^2

Answer

3(x^2 + 1)^2 * x

Answer

6x^2(x^2 + 1)

Question 7

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

Accepted Answer

16/3

Answer

14/3

Answer

28/3

Answer

32/3

Question 8

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

Accepted Answer

e^x + C

Answer

e^x - C

Answer

xe^x + C

Answer

xe^x - C

Question 9

Which of the following is the equation of the tangent line to the curve y = x^3 - 2x^2 + 1 at the point (1, 0)?

Accepted Answer

y = x - 1

Answer

y = x + 1

Answer

y = 2x - 1

Answer

y = -2x + 1

Question 10

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

Accepted Answer

y = x - 1

Answer

y = -x + 1

Answer

y = -2x + 1

Answer

y = 2x - 1

Question 11

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

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

2x - 2

Answer

x + 3

Question 12

Which technique is used to find the derivative of a logarithmic function?

Accepted Answer

The chain rule

Answer

The product rule

Answer

The quotient rule

Answer

Integration by parts

Question 13

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

Accepted Answer

2x/(x^2 + 1)

Answer

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

Answer

x/(x^2 + 1)

Answer

1/(x^2 + 1)

Question 14

What is the formula for the sum of an infinite geometric series?

Accepted Answer

S = a/(1 - r), where |r| < 1

Answer

S = a*(1 - r), where |r| < 1

Answer

S = a/r, where |r| < 1

Answer

S = a*r, where |r| < 1

Question 15

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

Accepted Answer

3x^2 - 4x + 5

Answer

x^3 - 4x + 5

Answer

3x^2 - 2x + 5

Answer

x^3 - 2x^2 + 5

Question 16

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x^2 - 5

Answer

3x^2 - 5

Answer

x^3 + 4x - 5

Question 17

Find the limit of (sin(x) - x)/x as x approaches 0.

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not exist

Question 18

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

Accepted Answer

y = x + 1

Answer

y = e^x

Answer

y = e^x - 1

Answer

y = x

Question 19

Find the indefinite integral of ∫(x/(x^2 + 1)) dx.

Accepted Answer

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

Answer

ln(x^2 + 1) + C

Answer

x^2 + 1 + C

Answer

x/(x^2 + 1) + C

Question 20

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

Accepted Answer

3x² + 4x - 5

Answer

3x² + 2x - 5

Answer

x³ + 4x² - 5

Answer

x³ + 2x² - 10

Question 21

Find the indefinite integral of 1/(x² + 1).

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

ln(x² + 1) + C

Answer

1/(x² + 1) + C

Question 22

What is the derivative of the function  *f(x) = x³ - 2x² + 5x - 7*?

Accepted Answer

3x² - 4x + 5

Answer

x² - 4x + 5

Answer

3x³ - 8x² + 15x - 21

Answer

x³ - 4x² + 10x - 14

Question 23

What is the indefinite integral of *∫(x² + 2x - 3) dx*?

Accepted Answer

(x³/3) + x² - 3x + C

Answer

x³ + 2x² - 3x + C

Answer

(x³/3) + x² + 3x + C

Answer

(x³/2) + 2x² - 3x + C

Question 24

Find the equation of the tangent line to the curve *y = x³ - 2x² + 1* at the point (1, 0).

Accepted Answer

y = 3x - 3

Answer

y = x² - 2x + 1

Answer

y = 3x + 2

Answer

y = x³ - 2x² - 1

Question 25

What is the area under the curve *y = sin(x)* between *x = 0* and *x = π*?

Accepted Answer

2

Answer

π/2

Answer

1

Answer

π

Question 26

Evaluate the limit as *x* approaches infinity of *(x² - 3x + 2)/(x² + 2x - 1)*.

Accepted Answer

1

Answer

-1

Answer

0

Answer

∞

Question 27

What is the volume of the solid generated by rotating the curve *y = x²* between *x = 0* and *x = 2* about the x-axis?

Accepted Answer

16π/3

Answer

64π/3

Answer

32π/3

Answer

8π/3

Question 28

Is the sequence defined by *a_n = n² - 3n + 2* convergent or divergent?

Accepted Answer

Divergent

Answer

Oscillating

Answer

Convergent

Question 29

What is the Taylor series expansion of *f(x) = e^x* about *x = 0*?

Accepted Answer

∑_(n=0)^∞ (x^n)/n!

Answer

∑_(n=0)^∞ (x^n)/(n+1)!

Answer

∑_(n=1)^∞ x^n

Answer

∑_(n=1)^∞ (x^n)/n

Question 30

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

Accepted Answer

f'(x) = 6x + 2

Answer

f'(x) = 3x + 2

Answer

f'(x) = 6x - 1

Answer

f'(x) = 3x^2

Question 31

What is the integral of the function f(x) = 2x + 1?

Accepted Answer

∫(2x + 1) dx = x^2 + x + C

Answer

∫(2x + 1) dx = 2x + C

Answer

∫(2x + 1) dx = x^2 + C

Answer

∫(2x + 1) dx = 2x^2 + x + C

Question 32

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

Accepted Answer

x = 0, x = 2

Answer

x = -1, x = 2

Answer

x = 1, x = 2

Answer

x = 0, x = 1

Question 33

The area under the curve y = x^2 from x = 0 to x = 2 is:

Accepted Answer

8/3

Answer

4

Answer

2

Answer

4/3

Question 34

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

Accepted Answer

y = 2x

Answer

y = 2x + 1

Answer

y = x + 2

Answer

y = x + 1

Question 35

A particle moves along a straight line with its position given by s(t) = t^3 - 6t^2 + 9t. Find the particle's velocity at time t = 2.

Accepted Answer

-3

Answer

3

Answer

-6

Answer

6

Question 36

The function f(x) = 1/x is:

Accepted Answer

Decreasing on its entire domain

Answer

Increasing on its entire domain

Answer

Increasing on (0, ∞) and decreasing on (-∞, 0)