AP Calculus AB (May 2025) Units 1-2. Include equations and laws!

Question 1

What is the definition of a limit?

Answer 1

A limit is the value that a function approaches as the input approaches some value.

Question 2

True or False: The limit of f(x) as x approaches c exists if f(x) approaches the same value from both the left and right.

Answer 2

True

Question 3

Fill in the blank: The limit notation is expressed as lim_x→c f(x) = L, where L is _____ .

Answer 3

the limit value

Question 4

What is the formal definition of the derivative?

Answer 4

The derivative of a function f at a point x is defined as

Question 5

What is the Power Rule for differentiation?

Answer 5

If f(x) = xⁿ, then f’(x) = nx^(n-1).

Question 6

True or False: The derivative of a constant is zero.

Answer 6

True

Question 7

What is the Product Rule for differentiation?

Answer 7

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

Question 8

What is the Quotient Rule for differentiation?

Answer 8

If u(x) and v(x) are functions, then

Question 9

What is the Chain Rule for differentiation?

Answer 9

If f(g(x)) is a composite function, then

Question 10

What is the limit of 1/x as x approaches 0?

Answer 10

The limit does not exist.

Question 11

Define continuity at a point.

Answer 11

A function f is continuous at x = c if

Question 12

True or False: A polynomial function is continuous everywhere.

Answer 12

True

Question 13

What is the Intermediate Value Theorem?

Answer 13

If f is continuous on [a, b] and N is a number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

Question 14

What is the definition of an asymptote?

Answer 14

An asymptote is a line that a graph approaches but never touches.

Question 15

What is the derivative of sin(x) ?

Answer 15

cos(x)

Question 16

What is the derivative of cos(x)?

Answer 16

-sin(x)

Question 17

What is the derivative of ( e^x )?

Answer 17

( e^x )

Question 18

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

Answer 18

1/x

Question 19

What is the limit of sin(x) as x approaches 0?

Answer 19

0

Question 20

What is the limit of ( tan(x) ) as x approaches pi/2 ?

Answer 20

The limit does not exist.

Question 21

What is the definition of an increasing function?

Answer 21

A function f is increasing on an interval if for any ( x₁ < x₂ ), ( f(x₁) < f(x₂) ).

Question 22

What is the definition of a decreasing function?

Answer 22

A function f is decreasing on an interval if for any ( x₁ < x₂ ), ( f(x_1) > f(x_2) ).

Question 23

What are critical points?

Answer 23

Critical points are points in the domain of a function where the derivative is zero or undefined.

Question 24

What is the Mean Value Theorem?

Answer 24

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

Question 25

True or False: An absolute maximum must occur at a critical point.

Answer 25

False

Question 26

What is the first derivative test?

Answer 26

The first derivative test is a method to determine if a critical point is a local maximum, local minimum, or neither based on the sign of the derivative before and after the point.

Question 27

What does the second derivative test determine?

Answer 27

The second derivative test determines the concavity of a function and can indicate local maxima or minima.

Question 28

If \( f''(x) > 0 \), what can be said about the function f?

Answer 28

The function f is concave up.

Question 29

If \( f''(x) < 0 \), what can be said about the function f?

Answer 29

The function f is concave down.

Question 30

What is the equation for a tangent line at a point \( (a, f(a)) \)?

Answer 30

The equation is \( y - f(a) = f'(a)(x - a) \).

Question 31

What is the definition of a definite integral?

Answer 31

A definite integral represents the signed area under the curve of a function f from a to b.

Question 32

What is the Fundamental Theorem of Calculus?

Answer 32

If f is continuous on [a, b], then \( \int_a^b f(x) dx = F(b) - F(a) \), where F is an antiderivative of f.

Question 33

What is the notation for an indefinite integral?

Answer 33

The notation is \( \int f(x) dx + C \), where C is the constant of integration.

Question 34

True or False: The integral of a constant k is \( kx + C \).

Answer 34

True

Question 35

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

Answer 35

\( \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \).

Question 36

What is the area under the curve from x = a to x = b for the function f(x)?

Answer 36

It is given by \( \int_a^b f(x) dx \).

Question 37

What does it mean for a function to be differentiable?

Answer 37

A function is differentiable at a point if it has a derivative at that point.

Question 38

What is L'Hôpital's Rule used for?

Answer 38

L'Hôpital's Rule is used to evaluate limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

Question 39

What is the derivative of \( \sqrt{x} \)?

Answer 39

\( \frac{1}{2\sqrt{x}} \)

Question 40

What is the integral of \( e^x \)?

Answer 40

\( e^x + C \)

Question 41

What is the integral of \( \sin(x) \)?

Answer 41

\( -\cos(x) + C \)

Question 42

What is the integral of \( \cos(x) \)?

Answer 42

\( \sin(x) + C \)

Question 43

What is the relationship between the derivative and the slope of the tangent line?

Answer 43

The derivative at a point gives the slope of the tangent line to the curve at that point.

Question 44

What is the limit of \( \frac{f(x+h) - f(x)}{h} \) as h approaches 0?

Answer 44

It is the definition of the derivative, f'(x).

Question 45

What does it mean for a function to be continuous?

Answer 45

A function is continuous if there are no breaks, jumps, or holes in its graph.

Question 46

What is a horizontal asymptote?

Answer 46

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity.

Question 47

What is a vertical asymptote?

Answer 47

A vertical asymptote is a vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

Question 48

True or False: Every continuous function has an inverse.

Answer 48

False

Question 49

What is the area of a region bounded by the x-axis and the curve f(x) from x = a to x = b?

Answer 49

It is given by the definite integral \( \int_a^b f(x) dx \).

Question 50

What is the derivative of a constant function f(x) = c?

Answer 50

The derivative is zero: \( f'(x) = 0 \).

Question 51

What is the relationship between the first derivative and increasing/decreasing behavior?

Answer 51

If f'(x) > 0, f is increasing; if f'(x) < 0, f is decreasing.

Question 52

What is the limit of a function f(x) as x approaches a value c?

Answer 52

It is the value that f(x) gets closer to as x gets closer to c.

Question 53

What is the derivative of \( an(x) \)?

Answer 53

\( sec^2(x) \)

Question 54

What is the integral of \( an(x) \)?

Answer 54

\( -\ln|\cos(x)| + C \)

Question 55

What is the formula for the area of a triangle under a curve?

Answer 55

The area is given by \( \int_a^b f(x) dx \) where f(x) is the height.

Question 56

What is the relationship between the second derivative and concavity?

Answer 56

If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.

Question 57

What is the derivative of \( x^3 \)?

Answer 57

\( 3x^2 \)

Question 58

What is the integral of \( x^2 \)?

Answer 58

\( \frac{x^3}{3} + C \)

Question 59

What is the limit of \( f(x) = \frac{1}{x} \) as x approaches infinity?

Answer 59

0

Question 60

What is the limit of \( f(x) = x^2 \) as x approaches 2?

Answer 60

4

Question 61

What is the derivative of \( ln(x) \)?

Answer 61

\( \frac{1}{x} \)

Question 62

What is the integral of a step function?

Answer 62

The integral of a step function is the sum of the areas of rectangles formed by the steps.