Master Calculus Derivatives: Enhance Your Mathematical Skills

Question 1

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

Accepted Answer

(A) 3x² + 4x - 5

Answer

(B) x³ + 4x² - 10x + 1

Answer

(C) 3x² + 2x - 5

Answer

(D) x² + 4x - 5

Question 2

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 - 5

Question 3

Which of the following is NOT a technique for finding the derivative of a function?

Accepted Answer

Riemann sum

Answer

Limit definition

Answer

Power rule

Answer

Product rule

Question 4

Which of the following functions has a constant derivative?

Accepted Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = ln(x)

Question 5

To demonstrate the power rule, use the limit definition of the derivative to prove that the derivative of f(x) = x^n is n * x^(n-1).

Accepted Answer

Apply the limit definition of the derivative to f(x) = x^n and simplify.

Answer

Use the product rule to differentiate f(x) = x^n.

Answer

The derivative of x^n is directly nx^(n-1) without any need for proof.

Answer

The derivative of x^n can be found using the quotient rule.

Question 6

Calculate the derivative of the function f(x) = x² + 2x + 1 using the limit definition.

Accepted Answer

2x + 2

Answer

2x + 1

Answer

x + 2

Answer

x + 1

Question 7

Determine the slope of the tangent line to the graph of f(x) = sin(x) at x = π/4.

Accepted Answer

√2/2

Answer

1

Answer

-1

Answer

-√2/2

Question 8

Which of the following functions has a vertical tangent at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x³

Answer

f(x) = cos(x)

Answer

f(x) = e^x

Question 9

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 10

Using the limit definition, find the derivative of the function g(x) = x^2 + 2x - 1.

Accepted Answer

2x + 2

Answer

x + 2

Answer

2x - 1

Answer

x - 1

Question 11

Find the equation of the tangent line to the graph of the function g(x) = cos(x) at x = π/2.

Accepted Answer

y = 0x + 1

Answer

y = -x + π/2

Answer

y = x - π/2 + 1

Answer

y = x + π/2

Question 12

Which of the following is a second-order derivative of the function f(x) = ln(x)?

Accepted Answer

f''(x)

Answer

f'(x)

Answer

f'''(x)

Answer

f''''(x)

Question 13

Find the derivative of the inverse function of f(x) = x^4 + 2.

Accepted Answer

1 / (4(x^4 + 2)^3)

Answer

1 / (4x^3)

Answer

4x^3

Answer

4(x^4 + 2)^3

Question 14

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

Accepted Answer

2x + 1

Answer

x^2 + 2

Answer

2x - 1

Answer

x + 2

Question 15

Find the critical points of the function f(x) = |x - 2|.

Accepted Answer

2

Answer

0

Answer

1

Answer

3

Question 16

The limit definition of the derivative is expressed as:

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 17

The derivative rule for exponential functions is:

Accepted Answer

d/dx (e^x) = e^x

Answer

d/dx (e^x) = x * e^(x-1)

Answer

d/dx (e^x) = x^2 * e^(x-2)

Answer

d/dx (e^x) = 1 / e^x

Question 18

Applying the chain rule, find the derivative of f(x) = 2^x:

Accepted Answer

2^x * ln(2)

Answer

2^(x-1)

Answer

2x^(x-1)

Answer

ln(2) / 2^x

Question 19

If f'(2) = 5, what is the slope of the tangent line to the graph of f(x) at x = 2?

Accepted Answer

5

Answer

2

Answer

7

Answer

10

Question 20

The limit definition of the derivative of a function f(x) is given by:

Accepted Answer

limh→0(f(x + h) - f(x - h))/2h

Answer

limh→0(f(x + h) - f(0))/h

Answer

limh→∞(f(x + h) + f(x - h))/2h

Answer

limh→∞(f(x + h) - f(x - h))/2h

Question 21

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

Accepted Answer

2x + 5

Answer

x^2 + 5

Answer

2x - 3

Answer

x + 5

Question 22

Which of the following rules can be used to find the derivative of f(x) = (x^3 - 1)^4?

Accepted Answer

Chain rule

Answer

Power rule

Answer

Product rule

Answer

Quotient rule

Question 23

Find the equation of the tangent line to the curve y = f(x) at the point (1, 4) if f'(1) = 2.

Accepted Answer

y = 2x + 2

Answer

y = 2x + 0

Answer

y = -2x + 6

Answer

y = -2x + 2

Question 24

Which of the following functions is not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 25

The limit definition of the derivative is given by:

Accepted Answer

lim(h→0) [f(x+h)-f(x)]/h

Answer

The average rate of change of the function

Answer

The slope of the normal line at a given point

Answer

The integral of the function

Question 26

What is the derivative of the function f(x) = e^(2x)?

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

2xe^(2x)

Answer

1/e^(2x)

Question 27

Find the slope of the tangent line to the curve y = x^2 + 2x - 1 at x = 2:

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 28

Find the absolute maximum value of the function f(x) = -x^2 + 6x - 8 on the interval [-2,4]:

Accepted Answer

7

Answer

3

Answer

11

Answer

15

Question 29

Which of the following is the second derivative of f(x) = cos(x)?

Accepted Answer

-cos(x)

Answer

cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 30

Which of the following is NOT a step in finding the derivative of a function f(x) using the chain rule?

Accepted Answer

Take the derivative of the product directly.

Answer

Find the derivative of the outer function.

Answer

Find the derivative of the inner function.

Answer

Multiply the derivatives together.

Question 31

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/(x^2 + 1)

Answer

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

Question 32

Utilize the limit definition to determine the derivative of the function f(x) = x^2 + 2x - 1.

Accepted Answer

2x + 2

Answer

x^2 + 2x - 1

Answer

2x - 2

Answer

x^2 - 2x + 1

Question 33

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

Accepted Answer

e^x

Answer

e^x - 1

Answer

e^x + 1

Answer

ln(e^x)

Question 34

Determine the second derivative of the function f(x) = sin(x).

Accepted Answer

-cos(x)

Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Question 35

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

Accepted Answer

x = 0, 2

Answer

x = -1, 1

Answer

x = 0, 1

Answer

x = -2, 2

Question 36

Which of the following is a prerequisite for a local minimum of a function?

Accepted Answer

Zero first derivative at that point

Answer

Positive second derivative at that point

Answer

Undefined first derivative at that point

Answer

Negative second derivative at that point

Question 37

Find the absolute maximum and minimum values of the function f(x) = -x^2 + 4x + 3 on the interval [0, 3].

Accepted Answer

Max: (2, 7); Min: (0, 3)

Answer

Max: (2, 7); Min: (3, 0)

Answer

Max: (3, 12); Min: (0, 3)

Answer

Max: (0, 3); Min: (2, 7)

Question 38

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

Accepted Answer

6xe^(3x^2 - 2x + 1)

Answer

e^(3x^2 - 2x + 1)

Answer

3e^(3x^2 - 2x + 1)

Answer

-2e^(3x^2 - 2x + 1)

Question 39

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

Accepted Answer

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

Answer

lim_(x->a) f'(x)

Answer

lim_(x->0) f'(x)

Answer

lim_(h->∞) [f(x+h) - f(x)]/h

Question 40

What is the slope of the tangent line to the curve y = f(x) at x = 2 if f'(2) = 5?

Accepted Answer

5

Answer

-5

Answer

2

Answer

10

Question 41

Which of the following is NOT a technique for finding derivatives?

Accepted Answer

Integration by parts

Answer

Power rule

Answer

Chain rule

Answer

Quotient rule

Question 42

Find the derivative of y = (x² + 1)(x - 3) using the product rule.

Accepted Answer

2x(x - 3) + (x² + 1)

Answer

2x(x² + 1) + (x - 3)

Answer

(x² + 1)(x - 3)'

Answer

(x² + 1)'(x - 3)'

Question 43

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

cos²(x)

Answer

sin(x)

Question 44

Which of the following represents the second derivative of f(x)?

Accepted Answer

d²f/dx²

Answer

dd/dxf

Answer

df/dx²

Answer

d²x/df

Question 45

A particle moves along the x-axis such that its position at time t is given by s(t) = t³ - 6t² + 9t. What is the particle's velocity at t = 2?

Accepted Answer

0

Answer

-6

Answer

6

Answer

2

Question 46

Find the critical points of the function f(x) = x⁴ - 4x².

Accepted Answer

x = 0, ±2

Answer

x = 0, ±1

Answer

x = 1, ±2

Answer

No critical points

Question 47

Which of the following is the equation of the tangent line to the curve y = f(x) at the point (2, 5) if f'(2) = 3?

Accepted Answer

y - 5 = 3(x - 2)

Answer

y - 3 = 5(x - 2)

Answer

y - 2 = 5(x - 3)

Answer

y - 2 = 3(x - 5)

Question 48

What is the derivative of the function f(x) = 3x² - 2x + 1?

Accepted Answer

6x - 2

Answer

9x - 2

Answer

6x² - 2

Answer

3x - 2

Question 49

What is the slope of the tangent line to the curve y = x³ at the point (2, 8)?

Accepted Answer

12

Answer

24

Answer

6

Answer

8

Question 50

Find the derivative of the function f(x) = sin(x) using the limit definition.

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-cos(x)

Answer

-sin(x)

Question 51

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

Accepted Answer

e^x

Answer

1/e^x

Answer

x*e^(x-1)

Answer

ln(x)

Question 52

Find the critical points of the function f(x) = x³ - 3x² + 2.

Accepted Answer

x = 0, x = 2

Answer

x = -1, x = 1

Answer

x = 0, x = 1

Answer

x = 1, x = 2

Question 53

The position of an object is given by s(t) = t² - 4t + 3. What is the object's velocity at time t = 2?

Accepted Answer

0

Answer

4

Answer

-2

Answer

2

Question 54

What is the derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

x

Answer

ln(x)/x

Answer

-1/x

Question 55

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

Accepted Answer

y = 2x

Answer

y = x + 1

Answer

y = 2x + 1

Answer

y = x + 2

Question 56

What is the derivative of the function f(x) = cos(x)?

Accepted Answer

-sin(x)

Answer

sin(x)

Answer

cos(x)

Answer

-cos(x)

Question 57

A rectangular garden is to be enclosed by a fence on three sides, with the house forming the fourth side. If 100 feet of fencing is available, what dimensions will maximize the garden's area?

Accepted Answer

25 feet by 50 feet

Answer

30 feet by 40 feet

Answer

20 feet by 60 feet

Answer

10 feet by 70 feet