Calculus 1 Online Quizzes: Master Differential Calculus Concepts

Question 1

When the limit of a function exists at a particular point, what can be inferred about the function's behavior at that point?

Accepted Answer

The function is continuous at that point.

Answer

The function is differentiable at that point.

Answer

The function has a zero value at that point.

Question 2

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

Accepted Answer

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

Answer

2x + 3

Answer

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

Question 3

Calculate the definite integral ∫[0, 4] (x^2 + 2x - 3) dx using the Fundamental Theorem of Calculus, Part 2.

Accepted Answer

20.5

Answer

10

Answer

30

Answer

40

Question 4

Define the concept of a limit in calculus.

Accepted Answer

A limit describes a function's behavior as the input approaches a particular value.

Answer

A limit is always equal to the function's value at the input.

Answer

A limit only exists for functions that are continuous.

Answer

A limit indicates how a function behaves as the input becomes infinitely large.

Question 5

Evaluate the integral ∫x^2sin(x)dx using integration by parts.

Accepted Answer

-x^2cos(x) + 2xsin(x) + C

Answer

x^2cos(x) + 2xsin(x) + C

Answer

x^2sin(x) + 2xcos(x) + C

Answer

-x^2sin(x) - 2xcos(x) + C

Question 6

Find the limit of the function f(x) = (x^2 - 1) / (x - 1) as x approaches 1.

Accepted Answer

1

Answer

0

Answer

2

Answer

Does not exist

Question 7

Define the limit of a function f(x) as x approaches a.

Accepted Answer

The value that f(x) approaches indefinitely as x gets closer to a

Answer

The value that f(a) approaches as x gets closer to a

Answer

The value that f(x) equals when x equals a

Answer

The value that f(a) equals when x equals a

Question 8

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

Accepted Answer

2x

Answer

x^2

Answer

2

Answer

1

Question 9

Precisely define the limit of a function.

Accepted Answer

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

Answer

The value of the function at a specific input value

Answer

The slope of the function at a specific input value

Answer

The area under the curve of the function between two input values

Question 10

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

Accepted Answer

2x

Answer

x^2

Answer

1 / x^2

Answer

0

Question 11

Find the area enclosed by the curves y = ln(x) and y = x² between x = 1 and x = e.

Accepted Answer

e - 1

Answer

1

Answer

e

Answer

ln(e) - 1

Question 12

What is the surface area of the curve formed by the function f(x) = x³ - 2x² + 1 between x = 0 and x = 2?

Accepted Answer

∫[0,2] √(1 + (f'(x))²) dx

Answer

2π

Answer

4π

Answer

∫[0,2] (x³ - 2x² + 1) dx

Question 13

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

Accepted Answer

3x^2 + 4x - 5

Answer

3x + 4

Answer

x^2 + 4x - 5

Answer

3x^3 + 4x^2

Question 14

Determine whether the following sequence converges or diverges: a_n = (n + 1)/(2n - 1).

Accepted Answer

Converges to 1/2

Answer

Converges to -1/2

Answer

Diverges to +∞

Answer

Diverges to -∞

Question 15

Find the sum of the following infinite series: ∑(n=1 to ∞) (1/2)^n.

Accepted Answer

1

Answer

1/2

Answer

2

Question 16

Determine the Taylor series expansion of the function f(x) = e^x about the point x = 0.

Accepted Answer

∑(n=0 to ∞) (x^n/n!)

Answer

∑(n=1 to ∞) (x^(n+1)/n!)

Answer

∑(n=0 to ∞) (x^(n+1)/(n+1)!)

Answer

∑(n=1 to ∞) (x^n/n)

Question 17

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 - 5x + 1

Answer

2x^2 + 4x - 5

Answer

3x + 2

Question 18

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

Accepted Answer

4

Answer

undefined

Answer

0

Answer

2

Question 19

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

Accepted Answer

y = -x + 1

Answer

y = -2x + 1

Answer

y = 2x - 1

Answer

y = x + 1

Question 20

Calculate the area under the curve y = 2x + 1 from x = 0 to x = 2.

Accepted Answer

6

Answer

3

Answer

12

Answer

8

Question 21

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

Accepted Answer

-1, 1

Answer

1, 2

Answer

0, 1

Answer

-1, 2

Question 22

Determine the inflection point(s) of the function f(x) = x^4 - 2x^2 + 1.

Accepted Answer

(0, 1)

Answer

None

Answer

(-1, 2)

Answer

(1, 0)

Question 23

Calculate the derivative of the inverse trigonometric function f(x) = sin^-1(x).

Accepted Answer

1 / sqrt(1 - x^2)

Answer

-1 / sqrt(1 - x^2)

Answer

sqrt(1 - x^2)

Answer

x / sqrt(1 - x^2)

Question 24

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

Accepted Answer

e^x + x^2 + C

Answer

e^x + 2 + C

Answer

x^2 + 2 + C