Calculus 1: Master Limits, Derivatives, and Integrals

Question 1

Find the limit as x approaches 0 of the function:
(sin(x) - x) / x^3

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit does not exist

Question 2

In integration, which technique involves replacing the independent variable with a new variable?

Accepted Answer

U-Substitution (Change of Variable)

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Logarithmic differentiation

Question 3

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 4

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

Accepted Answer

2x^2 + 5x - 3

Answer

2x + 3

Answer

2x - 1

Answer

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

Question 5

What is the derivative of the function f(x) = (x² + 3x - 2)⁵?

Accepted Answer

5(x² + 3x - 2)⁴ * (2x + 3)

Answer

(x² + 3x - 2)⁴

Answer

5x⁴ + 15x³ - 10x²

Question 6

Find the integral of (x^2 + 2x - 1) / (x-1) with respect to x.

Accepted Answer

(x + 3) + 2ln|x - 1| + C

Answer

x^2 + x + C

Answer

x + 2 + C

Answer

x^2 + 3x + C

Question 7

Which concept is NOT a fundamental topic in Calculus I?

Accepted Answer

Integration by parts

Answer

Limits of functions

Answer

Continuity of functions

Answer

Applications of the derivative in optimization

Question 8

Evaluate the improper integral ∫[1, ∞) (e^(-x) / x) dx and determine whether it is convergent or divergent. Explain your reasoning.

Accepted Answer

The integral converges to 1.

Answer

The integral diverges to infinity.

Answer

The integral diverges to negative infinity.

Question 9

Which of the following is an improper integral?

Accepted Answer

∫[0, ∞] e^-x dx

Answer

∫[0, 1] 1 / x dx

Answer

∫[-1, 1] x^2 dx

Answer

∫[-π, π] sin(x) dx

Question 10

Which of the following is a vector function?

Accepted Answer

f(x, y, z) = (x, y, z)

Answer

f(x) = x^2 + 2x

Answer

f(x, y) = sin(x) cos(y)

Answer

f(x, y, z) = x^2 + y^2 + z^2

Question 11

Which of the following vector fields is conservative?

Accepted Answer

F = (y, -x, 0)

Answer

F = (x, y, z)

Answer

F = (x^2, y^2, z^2)

Answer

F = (sin(x), cos(y), tan(z))

Question 12

Find the value of the improper integral: ∫[0,∞) e^(-x^2) dx

Accepted Answer

√(π/2)

Answer

0

Answer

∞

Answer

-∞

Question 13

Determine the volume of the solid generated by rotating the function f(x) = x^2 between x = 0 and x = 2 about the x-axis.

Accepted Answer

16π/5

Answer

32π/5

Answer

64π/5

Answer

128π/5

Question 14

Find 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 = x

Answer

y = 1

Question 15

Which of the following is the derivative of the function f(x) = 3x^2 - 4x + 1?

Accepted Answer

f'(x) = 6x - 4

Answer

f'(x) = 3x^2 - 4

Answer

f'(x) = 12x - 4

Answer

f'(x) = 9x^2 - 4x

Question 16

Which of the following statements about the function f(x) = 1 / x is true?

Accepted Answer

f(x) has a vertical asymptote at x = 0.

Answer

f(x) is continuous at x = 0.

Answer

f(x) is differentiable at x = 0.

Answer

f(x) has a horizontal asymptote at y = 1.

Question 17

Which of the following techniques is used to find the integral of the function f(x) = e^x * cos(x)?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Trig substitution

Answer

Partial fractions

Question 18

Which of the following is NOT a property of a continuous function?

Accepted Answer

Its derivative exists at every point in its domain.

Answer

Its graph can be drawn without lifting the pen.

Answer

Its limit exists at every point in its domain.

Answer

It has no gaps or "holes" in its graph.

Question 19

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

Accepted Answer

3x^2 - 4x + 1

Answer

3x^2 + 4x + 1

Answer

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

Answer

x^3 + 4x^2 + x + 1

Question 20

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

Accepted Answer

x = 0, ±1

Answer

x = 0, 1, 2

Answer

x = ±1, 2

Answer

The function has no critical points.

Question 21

Which statement accurately describes the derivative of a constant function?

Accepted Answer

Equal to zero

Answer

Equal to the constant

Answer

Equal to infinity

Answer

Undefined

Question 22

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

Accepted Answer

f'(x) = 3x² + 4x - 5

Answer

f'(x) = 2x² + 4x - 5

Answer

f'(x) = x² + 4x - 5

Answer

f'(x) = x² - 4x + 5

Question 23

Determine the limit of (x² + 1)/(x - 1) as x approaches 1.

Accepted Answer

Limit is 2

Answer

Limit is 0

Answer

Limit is infinity

Answer

Limit does not exist

Question 24

Which integral qualifies as an improper integral?

Accepted Answer

∫[0, ∞) e^(-x) dx

Answer

∫[-1, 1] sin(x) dx

Answer

∫[0, 1] x² dx

Answer

∫[π/2, π] cos(x) dx

Question 25

What is the definition of the limit of a function as x approaches a certain value c?

Accepted Answer

The value that the function approaches as x gets closer and closer to c

Answer

The value that the function equals at x = c

Answer

The rate of change of the function at x = c

Answer

The area under the curve of the function from x = c to x = infinity

Question 26

For a function to be continuous at a point, which condition must hold true?

Accepted Answer

The function must have a limit at that point

Answer

The function must be differentiable at that point

Answer

The function must be monotonic at that point

Answer

The function must be bounded at that point

Question 27

State the power rule for integrals.

Accepted Answer

∫x^n dx = x^(n+1) / (n+1) + C

Answer

∫x^n dx = nx^(n-1) + C

Answer

∫x^n dx = x^(n+1) + C

Answer

∫x^n dx = nx^(n+1) + C

Question 28

Identify a second order linear differential equation.

Accepted Answer

y'' + 2y' + y = 0

Answer

y' + 2y = 0

Answer

y''' + y' = 0

Answer

y'''' + y' = 0

Question 29

Consider the harmonic series $sum_{n=1}^infty rac{1}{n}$. Explain why both the ratio test and the integral test fail to determine whether this series converges or diverges.

Accepted Answer

The ratio test gives $lim_{n	oinfty} left| rac{a_{n+1}}{a_n} ight| = 1$, which is inconclusive.

Accepted Answer

The integral test cannot be applied because the function $f(x) = rac{1}{x}$ is not decreasing on the interval $[1, infty)$.

Answer

The alternating nature of the harmonic series does not affect the applicability of the ratio or integral tests.

Question 30

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

Accepted Answer

3x² + 4x

Answer

x³ + 4x²

Answer

6x + 4

Answer

x³ - 4x

Question 31

State the formula for the chain rule when finding the derivative of a composite function, f(g(x)).

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(g(x)) * d/dx[g(x)]

Answer

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

Question 32

Calculate the limit of the expression (cos(x) - 1)/x as x approaches 0.

Accepted Answer

0

Answer

Undefined

Answer

1

Answer

-1

Question 33

State the product rule formula for finding the derivative of the product of two functions, f(x) and g(x).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 34

Determine the derivative of the function f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x)

Answer

2xe^(2x + 1)

Answer

e^(2x) + 1

Question 35

Determine if the function h(x) = x^3 - 2x^2 + 1 is continuous at x = 1.

Accepted Answer

Yes

Answer

Only if x ≠ 1

Answer

Cannot be determined

Answer

No

Question 36

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

Accepted Answer

2

Answer

1

Answer

-2

Answer

0

Question 37

Determine if the following improper integral converges or diverges:
∫[0, ∞) 1/(x^2 + 1) dx

Accepted Answer

Converges

Answer

Cannot be determined

Answer

Diverges

Question 38

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

Accepted Answer

f'(x) = e^x + 1 / x

Answer

f'(x) = e^x * x

Answer

f'(x) = (e^x * ln(x)) / x

Answer

f'(x) = e^x - 1 / x

Question 39

Which of the following functions is differentiable but not continuous?

Accepted Answer

f(x) = |x|

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Question 40

Compute the second derivative of the function f(x) = e^(2x - 1).

Accepted Answer

4e^(2x - 1)

Answer

e^(2x - 1)

Answer

2e^(2x - 1)

Answer

0

Question 41

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

Accepted Answer

3x^2 + 4x

Answer

3x^3 - 5

Answer

x^2 + 4x

Answer

x^3 + 2x

Question 42

Which test can be used to determine if a function is continuous at a specific point?

Accepted Answer

Epsilon-delta definition

Answer

Rolle's Theorem

Answer

Mean Value Theorem

Answer

Fundamental Theorem of Calculus

Question 43

What condition must a function satisfy to have an absolute maximum or minimum?

Accepted Answer

First derivative is zero and second derivative changes sign.

Answer

First derivative is non-zero and second derivative is negative.

Answer

First derivative is non-zero and second derivative is positive.

Question 44

Find the indefinite integral of the function f(x) = e^x + x^2.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 45

What is the formula for the sum of the first n natural numbers?

Accepted Answer

n(n+1)/2

Answer

n(n-1)/2

Answer

(n+1)^2/2

Answer

n^2/2

Question 46

Write out the Taylor series expansion of the function f(x) = e^x centered at x = 0.

Accepted Answer

∑(n=0 to infinity) (x^n/n!)

Answer

∑(n=1 to infinity) (x^n/n)

Answer

∑(n=0 to infinity) (x^n)

Answer

∑(n=1 to infinity) (x^n/n!)

Question 47

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

Accepted Answer

f'(x) = 3x^2 + 2

Answer

f'(x) = 2x

Answer

f'(x) = x^2 + 2

Question 48

Which integral represents the area under the curve of f(x) = 2x + 1 over the interval [0, 2]?

Accepted Answer

∫[0, 2] (2x + 1) dx

Answer

2∫[0, 2] x dx + ∫[0, 2] dx

Answer

(2x + 1) ∫[0, 2] dx

Answer

∫[2, 0] (2x + 1) dx

Question 49

Determine if the function f(x) = x^2 + 1 is continuous at x = 0.

Accepted Answer

Yes, f(x) is continuous at x = 0.

Answer

No, f(x) is discontinuous at x = 0 because it has a different left-hand and right-hand limit.

Answer

f(x) is only continuous at x = 0 if it is differentiable there.

Answer

No, f(x) is discontinuous at x = 0 because it is not defined there.

Question 50

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

Accepted Answer

e^x + C

Answer

e^(x^2) + C

Answer

1 / e^x + C

Answer

x * e^x + C

Question 51

Determine if f(x) = |x| is differentiable at x = 0.

Accepted Answer

No, f(x) is not differentiable at x = 0.

Answer

f(x) is differentiable at x = 0 and its derivative is -1.

Answer

f(x) is differentiable at x = 0 and its derivative is 1.

Answer

f(x) is differentiable at x = 0 if and only if it is continuous there.

Question 52

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x^2 - 10x + 1

Answer

x^3 + 2x^2 - 5

Answer

3x^2 - 4x - 5

Question 53

Determine whether the function f(x) = x^4 - 2x^2 + 1 is continuous at x = 0.

Accepted Answer

Yes

Answer

Only if x > 0

Answer

No

Answer

Cannot be determined

Question 54

Calculate the area between the curves y = x^2 and y = 2x from x = 0 to x = 2.

Accepted Answer

8/3

Answer

2

Answer

4/3

Answer

16/3

Question 55

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

Accepted Answer

π/5

Answer

π/10

Answer

2π/5

Answer

4π/5

Question 56

Determine whether the function f(x) = x^3 - 2x^2 + 1 is continuous at x = 0.

Accepted Answer

Yes

Answer

Cannot be determined

Answer

No

Question 57

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

Accepted Answer

**Simplified:** 3x^2 - 4x + 5

Answer

x^3 - 4x^2 + 10x - 1

Answer

3x^3 - 4x^2 + 10x

Answer

x^2 - 4x + 5

Question 58

Evaluate the integral of sin(x) dx from x = 0 to x = π/2.

Accepted Answer

**Corrected:** 1

Answer

π/2

Answer

0

Answer

2

Question 59

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

Accepted Answer

**Simplified:** 8/3

Answer

16/3

Answer

2

Question 60

Determine the convergence or divergence of the series ∑(n=1 to ∞) 1/n^2.

Accepted Answer

**Simplified:** Convergent

Answer

Cannot be determined

Answer

Conditionally convergent

Answer

Divergent

Question 61

Find the Taylor series expansion of f(x) = e^x about x = 0.

Accepted Answer

**Corrected:** ∑(n=0 to ∞) (x^n / n!)

Answer

∑(n=1 to ∞) (x^n / n)

Answer

∑(n=1 to ∞) x^n

Answer

∑(n=0 to ∞) (x^n / 2^n)

Question 62

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

Accepted Answer

π/2

Answer

π/4

Answer

π

Answer

2π

Question 63

A particle moves along a straight line with position function s(t) = t^3 - 6t^2 + 9t + 2. Find the acceleration of the particle at time t = 2. Show your work.

Accepted Answer

6

Answer

0

Answer

-6

Answer

12

Question 64

What is the derivative of the function f(x) = x^3 + 2x^2 - 5?

Accepted Answer

3x^2 + 4x

Answer

x^2 + 4x

Answer

3x^2 - 4x

Answer

x^3 + 4x