Calculus Basics

Question 1

What is the derivative of a constant?

Answer 1

0

Question 2

What is the power rule for differentiation?

Answer 2

If f(x) = x^n, then f’(x) = n*x^(n-1)

Question 3

True or False: The derivative of sin(x) is cos(x).

Answer 3

True

Question 4

What is the integral of x^n with respect to x?

Answer 4

(x^(n+1))/(n+1) + C, where n ≠ -1

Question 5

Fill in the blank: The limit of a function as x approaches a point must exist for the function to be ________ at that point.

Answer 5

continuous

Question 6

What is the derivative of e^x?

Answer 6

e^x

Question 7

What is the integral of cos(x) with respect to x?

Answer 7

sin(x) + C

Question 8

What does it mean for a function to be continuous?

Answer 8

A function is continuous if it is defined at a point, the limit exists, and the limit equals the function’s value at that point.

Question 9

True or False: The sum rule of differentiation states that the derivative of a sum is the sum of the derivatives.

Answer 9

True

Question 10

What is the chain rule in differentiation?

Answer 10

If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x)

Question 11

What is the integral of 1/x with respect to x?

Answer 11

ln|x| + C

Question 12

What is the derivative of tan(x)?

Answer 12

sec^2(x)

Question 13

Fill in the blank: The Fundamental Theorem of Calculus connects ________ and ________.

Answer 13

differentiation, integration

Question 14

What is the product rule in differentiation?

Answer 14

If u and v are functions, then (uv)’ = u’v + uv’.

Question 15

True or False: The derivative of ln(x) is 1/x.

Answer 15

True

Question 16

What is the integral of sin(x) with respect to x?

Answer 16

-cos(x) + C

Question 17

What is the quotient rule in differentiation?

Answer 17

If u and v are functions, then (u/v)’ = (u’v - uv’)/v^2.

Question 18

What does the mean value theorem state?

Answer 18

If a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f’(c) = (f(b) - f(a))/(b - a).

Question 19

What is the second derivative test used for?

Answer 19

To determine the concavity of a function and to identify local maxima and minima.

Question 20

What is the integral of sec(x)tan(x) with respect to x?

Answer 20

sec(x) + C

Question 21

True or False: The limit of a function can be infinite.

Answer 21

True

Question 22

What is the derivative of x^3?

Answer 22

3x^2

Question 23

Fill in the blank: The derivative represents the ________ of a function at a point.

Answer 23

slope

Question 24

What is the integral of e^(ax) with respect to x?

Answer 24

(1/a)e^(ax) + C

Question 25

What is the derivative of a sum of functions?

Answer 25

The derivative of the sum is the sum of the derivatives.

Question 26

What is the integral of a constant k with respect to x?

Answer 26

kx + C

Question 27

True or False: The derivative of a product of two functions requires the use of the product rule.

Answer 27

True

Question 28

What is the limit definition of the derivative?

Answer 28

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

Question 29

What does the term 'critical point' refer to?

Answer 29

A critical point is where the derivative is zero or undefined.

Question 30

What is the integral of (1 + x^2) with respect to x?

Answer 30

x + (1/3)x^3 + C

Question 31

What is the derivative of sin(x^2)?

Answer 31

2x cos(x^2)

Question 32

Fill in the blank: A function is ________ if it has a derivative at every point in its domain.

Answer 32

differentiable

Question 33

What is the integral of tan(x) with respect to x?

Answer 33

-ln|cos(x)| + C

Question 34

True or False: The limit of a function at a point can be found using substitution.

Answer 34

True

Question 35

What is the derivative of cos(x)?

Answer 35

-sin(x)

Question 36

What is the integral of a constant multiplied by a function?

Answer 36

k * ∫f(x)dx = k * F(x) + C

Question 37

What is the difference between definite and indefinite integrals?

Answer 37

Definite integrals have limits and yield a numerical value, while indefinite integrals do not have limits and yield a function.

Question 38

What is the derivative of ln(x^2)?

Answer 38

2/x

Question 39

What is the integral of (ax + b)^n with respect to x?

Answer 39

(1/(a(n+1)))(ax + b)^(n+1) + C

Question 40

True or False: The second derivative of a function gives information about its concavity.

Answer 40

True

Question 41

What is the derivative of the inverse function f^-1(x)?

Answer 41

If y = f(x), then dy/dx = 1/(f'(f^-1(y)))

Question 42

What is the integral of sin^2(x) with respect to x?

Answer 42

(1/2)(x - sin(x)cos(x)) + C

Question 43

Fill in the blank: A function is ________ at a point if it can be drawn without lifting the pencil.

Answer 43

continuous

Question 44

What is the derivative of the exponential function a^x?

Answer 44

a^x ln(a)

Question 45

What is the integral of cos^2(x) with respect to x?

Answer 45

(1/2)(x + sin(x)cos(x)) + C

Question 46

Derivatives of arctan, arcsin, arccos.

Answer 46

(a/a^2+x^2) for arctan, 1/root(a^2-x^2) for arcsin, -1/root(a^2-x^2) for arccos

Question 47

Derivatives of arctan, arcsin, arccos.

Answer 47

(a/a^2+x^2) for arctan, 1/root(a^2-x^2) for arcsin, -1/root(a^2-x^2) for arccos