Calculus 1: Master the Fundamentals of Differential and Integral Calculus

Question 1

Find the derivative of the function defined implicitly by the equation xy^2 + x^3 = y.

Accepted Answer

dy/dx = (y^2 - 3x^2) / (2xy + 1)

Answer

dy/dx = (2xy + 1) / (3x^2 - y^2)

Answer

dy/dx = y / x

Answer

dy/dx = -y / x

Question 2

Find the partial derivatives of f(x,y) = x^2y + y^3 and explain their geometrical interpretation as the slope of tangent lines.

Accepted Answer

∂f/∂x = 2xy, ∂f/∂y = x^2 + 3y^2. The slope of the tangent line to the surface z = f(x, y) in the x-direction is 2xy, and in the y-direction is x^2 + 3y^2.

Answer

∂f/∂x = x^2, ∂f/∂y = y^3. These are the total derivatives, not the partial derivatives.

Answer

∂f/∂x = y, ∂f/∂y = x. These are the partial derivatives of f(x,y) = xy, not f(x,y) = x^2y + y^3.

Answer

The function has no partial derivatives.

Question 3

Determine the nth term and convergence or divergence of the series: ∑(n=1,∞) 1/(2^n + 3).

Accepted Answer

nth term: 1/(2^n + 3); convergent

Answer

nth term: 1/2^n; divergent

Answer

nth term: (2^n + 3)/n; divergent

Answer

nth term: (2^n + 3)/n; convergent

Question 4

Evaluate the improper integral ∫[0, ∞) e^(-x^2) dx. Does it converge or diverge?

Accepted Answer

Converges

Answer

Diverges to infinity

Answer

Diverges to negative infinity

Question 5

Evaluate the improper integral ∫[1, ∞) e^(-2x) dx. Which of the following statements about the improper integral is correct?

Accepted Answer

The improper integral converges to 1/2.

Answer

The improper integral diverges.

Answer

The improper integral converges to e^(-2).

Question 6

A city's population growth rate is directly proportional to the square of its population at any given time. Write the differential equation that models this scenario and solve it.

Accepted Answer

Differential equation: $rac{dp}{dt} = kp^2$, Solution: $p(t) = rac{1}{k(t-C)}$

Answer

Differential equation: $rac{dp}{dt} = k$, Solution: $p(t) = Ce^{kt}$

Answer

Differential equation: $rac{dp}{dt} = kp$, Solution: $p(t) = Ce^{kt}$

Answer

Differential equation: $rac{dp}{dt} = ksqrt{p}$, Solution: $p(t) = C^2e^{kt}$

Question 7

Calculate the partial derivative of f(x, y) = x^2 * y + y^3 with respect to x.

Accepted Answer

2xy + 3y^2

Answer

x^2 + 3y^2

Answer

2x + 3y^2

Answer

x^2 + 3y

Question 8

Calculate the derivative of f(x) = (x^2 + 3x - 2) / (x - 1) using the quotient rule.

Accepted Answer

(x^2 + 6x - 5) / (x - 1)^2

Answer

(2x + 3) / (x - 1)

Answer

(2x^2 + 5x - 3) / (x - 1)^2

Answer

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

Question 9

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

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x - 5

Answer

3x² + 2x - 5

Question 10

Calculate the integral of ∫(4x² + 3x - 5) dx.

Accepted Answer

4x³/3 + 3x²/2 - 5x + C

Answer

x³ + 3x² - 5x + C

Answer

3x³ + 4x² - 5x + C

Answer

12x² + 9x - 5 + C

Question 11

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

Accepted Answer

The rate at which the function changes with respect to x

Answer

The area under the curve of the function

Answer

The slope of the tangent line to the graph of the function at a given x value

Answer

The integral of the function

Question 12

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

Accepted Answer

∫[0,2] (2x + 1) dx

Answer

∫[2,0] (2x + 1) dx

Answer

∫[0,2] (2 + x) dx

Answer

∫[0,2] (2 - x) dx

Question 13

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

Accepted Answer

1/x + e^x

Answer

1/x - e^x

Answer

x + e^x

Answer

x - e^x

Question 14

Find 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 + 2x - 5

Answer

x^3 + 2x^2 - 5

Question 15

Calculate the integral of the function (∫(2x + 5) dx).

Accepted Answer

x^2 + 5x + C

Answer

2x^2 + 5 + C

Answer

2x^2 + 5x

Answer

x^2 + 5

Question 16

Determine whether the following series converges or diverges: (sum_{n=1}^infty rac{1}{n^2}).

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges conditionally

Question 17

Calculate the derivative of the function f(x) = sin(2x) + ln(x)

Accepted Answer

2cos(2x) + 1/x

Answer

2cos(2x) - 1/x

Answer

cos(2x) + 1/x

Answer

cos(2x) - 1/x

Question 18

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 19

Determine the concavity of the function f(x) = x^4 - 2x^2 + 1

Accepted Answer

Concave up for all x

Answer

Concave up for x < 0, concave down for x > 0

Answer

Concave down for all x

Answer

Concave down for x < 0, concave up for x > 0

Question 20

Use L'Hôpital's rule to find the limit of the function as x approaches 0: lim (e^x - 1) / x

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 21

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

Accepted Answer

3x^2 - 4x

Answer

3x^2 + 4x

Answer

2x - 1

Answer

5

Question 22

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

Accepted Answer

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

Answer

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

Answer

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

Answer

x^2 + 3x - 2 + C

Question 23

Find the limit of the function lim (x->2) (x^2 - 4) / (x - 2).

Accepted Answer

0

Answer

2

Answer

4

Answer

Does not exist

Question 24

Use the Mean Value Theorem to find a c in the interval [1, 3] such that f'(c) = (f(3) - f(1)) / (3 - 1) for the function f(x) = x^2.

Accepted Answer

c = 2

Answer

c = 1

Answer

c = 3

Answer

The Mean Value Theorem cannot be applied to this function.

Question 25

Determine whether the sequence {n / (n + 1)} is convergent or divergent.

Accepted Answer

Convergent to 0

Answer

Convergent to 1

Answer

Divergent to infinity

Answer

Divergent to negative infinity

Question 26

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

Accepted Answer

3x² - 2

Answer

3x

Answer

x³ - 2x²

Answer

x³ + 2x

Question 27

Evaluate the integral of ∫(2x + 5) dx.

Accepted Answer

x² + 5x + C

Answer

2x³ + 5x + C

Answer

x³ + 5x² + C

Answer

2x² + 5 + C

Question 28

Use the Chain Rule to find the derivative of f(x) = (3x² + 2)⁵.

Accepted Answer

30x(3x² + 2)⁴

Answer

5(3x² + 2)⁴

Answer

15x(3x² + 2)⁴

Answer

15(3x² + 2)³

Question 29

Which of the following is a definite integral?

Accepted Answer

∫(sin(x)) dx from 0 to π

Answer

∫(x²) dx

Answer

∫(1/x²) dx

Answer

∫(e^(-x)) dx

Question 30

Determine the equation of the tangent line to the curve y = sin(x) at the point (π/4, √2/2).

Accepted Answer

y = √2/2 + √2/2(x - π/4)

Answer

y = √2/2

Answer

y = √2/2 - √2/2(x - π/4)

Answer

y = -√2/2 + √2/2(x - π/4)

Question 31

Which of the following is the correct definition of the definite integral?

Accepted Answer

The area under the curve of a function over a given interval

Answer

The rate of change of a function with respect to a variable

Answer

The slope of the tangent line to a curve at a given point

Answer

The limit of a sum of products as the number of partitions approaches infinity

Question 32

Determine whether the series ∑(n=1 to ∞) (1 / n²) converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 33

Find the derivative of the function f(x) = (x³ - 2x² + 1) using the power rule and subtraction rule.

Accepted Answer

3x² - 4x

Answer

3x² + 4x

Answer

x² - 4x

Answer

x² + 4x

Question 34

State the Chain Rule for calculating derivatives of composite functions.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 35

State the Test for Increasing and Decreasing Functions, which relates the derivative to the function's behavior.

Accepted Answer

If f'(x) > 0, then f(x) is increasing.

Answer

If f'(x) < 0, then f(x) is increasing.

Answer

If f''(x) > 0, then f(x) is increasing.

Answer

If f''(x) < 0, then f(x) is increasing.

Question 36

State the Mean Value Theorem, which provides a relationship between the average and instantaneous rates of change of a function.

Accepted Answer

f'(c) = (f(b) - f(a))/(b - a)

Answer

f(c) = f(a) + f(b)/2

Answer

f''(c) = (f(b) - f(a))/(b - a)^2

Answer

∫[f(x)dx] from a to b = f(c)(b - a)

Question 37

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

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = 1, x = 2

Answer

x = 1

Answer

The function has no critical points

Question 38

Which theorem is used to calculate the area under a curve?

Accepted Answer

Fundamental Theorem of Calculus

Answer

Chain Rule

Answer

Product Rule

Answer

Quotient Rule

Question 39

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

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 40

Find the local minimum of the function f(x) = x^4 - 4x^2 + 3:

Accepted Answer

x = 1

Answer

x = 0

Answer

x = -1

Answer

The function has no local minimum

Question 41

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

Accepted Answer

3x² + 4x - 5

Answer

x² + 4x - 5

Answer

3x² - 4x + 5

Answer

x² - 4x + 5

Question 42

Identify the antiderivative of the function f(x) = cos(x).

Accepted Answer

sin(x) + C

Answer

sin(x)

Answer

cos(x) + C

Answer

-sin(x)

Question 43

Which of the following statements accurately describes the function f(x) = |x - 1|?

Accepted Answer

It is continuous but not differentiable at x = 1.

Answer

It is differentiable at x = 1.

Answer

It is neither continuous nor differentiable at x = 1.

Answer

It is both continuous and differentiable at x = 1.

Question 44

Calculate the volume of the solid generated by rotating the region bounded by the curves y = x and y = x² about the y-axis.

Accepted Answer

2π/3

Answer

π/6

Answer

π/3

Answer

π

Question 45

Find the derivative of the function f(x) = ln(x² + 1).

Accepted Answer

2x / (x² + 1)

Answer

1 / (x² + 1)

Answer

(2x) / (x² - 1)

Answer

1 / 2x

Question 46

Which of the following is the chain rule for derivatives?

Accepted Answer

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

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)

Question 47

What is the derivative of the inverse trigonometric function arctan(x)?

Accepted Answer

1 / (1 + x²)

Answer

-1 / (1 + x²)

Answer

x² / (1 + x²)

Answer

-x² / (1 + x²)

Question 48

Find the area under the curve of the function f(x) = x² from x = 0 to x = 2.

Accepted Answer

8/3

Answer

4/3

Answer

16/3

Answer

32/3

Question 49

Find the volume of the solid generated by rotating the region bounded by the curves y = x² and y = 4 about the x-axis.

Accepted Answer

32π

Answer

16π

Answer

64π

Answer

128π

Question 50

Use the limit definition of the derivative to find f'(x) for f(x) = x^3 - 2x^2 + 1.

Accepted Answer

3x^2 - 4x

Answer

3x^2 + 4x

Answer

x^3 - 4x^2 + 1

Answer

x^3 + 4x^2 - 1

Question 51

Find the slope of the tangent line to the curve y = sin(x) at x = π/4 using the definition of the derivative.

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

-√2/2

Question 52

Calculate the integral ∫(x^2 + e^x) dx.

Accepted Answer

x^3/3 + e^x + C

Answer

x^3/3 - e^x + C

Answer

x^2/2 + e^x + C

Answer

x^2/2 - e^x + C

Question 53

Find the antiderivative of f(x) = 1/(x^2 + 1).

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

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

Answer

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

Question 54

Use implicit differentiation to determine dy/dx for the equation x^2 + y^2 = 25.

Accepted Answer

-y/x

Answer

y/x

Answer

x/y

Answer

-x/y

Question 55

Find the critical points of the function f(x) = x⁴ - 4x² + 3 by finding its derivative, setting it equal to zero, and solving for x.

Accepted Answer

x = ±1, x = ±√3

Answer

x = ±1, x = 0

Answer

x = 0, x = ±2

Answer

x = 0, x = ±√3

Question 56

Which of the following is the correct formula for the chain rule? The chain rule allows us to find the derivative of a composite function, a function within a function.

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

Answer

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

Question 57

Find the indefinite integral of the function g(x) = e^x + x² using the integrals of exponential and polynomial functions.

Accepted Answer

e^x + (x^3)/3 + C

Answer

(x^3)/3 + e^x + C

Answer

e^x - (x^3)/3 + C

Answer

e^x - (x^3)/3 - C

Question 58

Determine the derivative of the logarithmic function f(x) = ln(x² + 1)

Accepted Answer

2x/(x² + 1)

Answer

1/(x² + 1)

Answer

x/(x² + 1)

Answer

-2x/(x² + 1)

Question 59

Calculate the limit of the function as x approaches 0: lim⁡(x→0) (e^x - 1) / x

Accepted Answer

1

Answer

0

Answer

∞

Answer

-1

Question 60

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

Accepted Answer

(6x^4 + 27x^3 - 14x^2 - 20x + 4)

Answer

(9x^4 + 21x^3 - 10x^2 - 10x + 4)

Answer

(6x^4 + 12x^3 - 5x^2 - 20x + 4)

Answer

(3x^4 + 7x^3 - 2x^2 - 8x + 2)

Question 61

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

Accepted Answer

1

Answer

0

Answer

2

Answer

π/2

Question 62

Determine the convergence or divergence of the improper integral ∫[1, ∞) 1/x^2 dx.

Accepted Answer

Converges

Answer

Diverges

Answer

Conditional convergence

Question 63

A particle's position is given by the function *s(t) = t³ - 6t² + 9t*, where *t* is time in seconds. At what time(s) is the particle's velocity zero?

Accepted Answer

t = 1 second and t = 3 seconds

Answer

t = 0 seconds

Answer

t = 2 seconds

Answer

t = 9 seconds

Question 64

What is the derivative of the function *f(x) = 5x⁴ - 2x³ + 7*?

Accepted Answer

f'(x) = 20x³ - 6x²

Answer

f'(x) = 5x³ - 2x²

Answer

f'(x) = 9x³ - 5x²

Answer

f'(x) = 20x³ - 6x² + 7