AP Calculus AB Chapter 2 Review

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Questions and Answers

What is a continuous function?

Any function where the graph can be sketched in one continuous motion without lifting the pencil (correct)

A function that has at least one discontinuity

Any function that is defined for all real numbers

Any function that is polynomial

A function y=f(x) is continuous at an interior point c in its domain if...

Limit of f(x) as x approaches c = f(c)

A function y=f(x) is continuous at a left endpoint a of its domain if...

Limits of f(x) as x approaches a from the right = f(a)

A function y=f(x) is continuous at a right endpoint b of its domain if...

Limits of f(x) as x approaches b from the left = f(b) Signup and view all the answers

A continuous function is...

Continuous at every point in its domain Signup and view all the answers

All composites of continuous functions are...

Continuous Signup and view all the answers

What are the two predominant types of discontinuity?

Jump discontinuity & Removable discontinuity Signup and view all the answers

What does the Intermediate Value Theorem state?

A function y=f(x) that is continuous on a closed interval [a,b] takes every value between f(a) and f(b) Signup and view all the answers

Jump discontinuities occur where...

The graph has a break in it Signup and view all the answers

Removable discontinuity occurs at a point where...

There is a hole in the graph, which is where the value causes both the numerator and denominator of the function to be equal to zero Signup and view all the answers

Find the points at which the function f is continuous and the points at which f is discontinuous for the function f(x)=sqrt(2x+3).

Continuous: [-3/2, Infinity); Discontinuous: (-Infinity, -3/2) Signup and view all the answers

Give a formula for the extended function g that is continuous at the indicated point f(x)= (x^2-9)/(x+3) at x=-3.

g(x)=x-3 Signup and view all the answers

What is the average rate of change?

Change in distance divided by change in time Signup and view all the answers

Find the average speed during the first 3 seconds of a fall represented by 16t^2.

48 ft/s Signup and view all the answers

What is the instantaneous rate of change?

The speed at one instance of time Signup and view all the answers

What is lim x->0 sinx/x?

1 Signup and view all the answers

What does the Sum Rule state?

lim x->c (f(x)+g(x)) = sum of limits separately Signup and view all the answers

What does the Difference Rule state?

lim x->c (f(x)-g(x)) = difference of limits separately Signup and view all the answers

What does the Product Rule state?

lim x->c (f(x)*g(x)) = product of limits separately Signup and view all the answers

What does the Constant Multiple Rule state?

lim x->c (kf(x)) = klim x->c f(x) Signup and view all the answers

What does the Quotient Rule state?

lim x->c (f(x)/g(x)) = quotient of limits separately (bottom can't equal zero) Signup and view all the answers

What does the Power Rule state?

lim x->c f(x)^t = answer^t Signup and view all the answers

What is a Right Hand Limit?

Lim x->c+ f(x) Signup and view all the answers

What is a Left Hand Limit?

Lim x->c- f(x) Signup and view all the answers

What is a Two Sided Limit?

Lim x->c f(x) Signup and view all the answers

What is the Squeeze Theorem?

If G(x) is less than or equal to f(x), which is less than or equal to H(x) for all x not equal to c, and lim x->c G(x) = lim x->c H(x) = L, then lim x->c f(x) = L Signup and view all the answers

What is lim x->c (2x^3-3x^2+x-1)?

2c^3-3c^2+c-1 Signup and view all the answers

What is lim x->1 (x^3+3x^2-2x-17)?

-15 Signup and view all the answers

What is the limit x->0 tanx/x?

1 Signup and view all the answers

What is the Sum Rule of Limits?

If the limit as x approaches plus or minus infinity of f(x) = L and the limit as x approaches plus or minus infinity of g(x) = M, then the limit as x approaches plus or minus infinity of (f(x)+g(x)) = L+M Signup and view all the answers

What is the Difference Rule of Limits?

What is the Product Rule of Limits?

What is the Quotient Rule of Limits?

Study Notes

Continuous Functions

A continuous function can be graphed in one motion, without lifting the pencil.
A function ( y = f(x) ) is continuous at an interior point ( c ) if ( \lim_{x \to c} f(x) = f(c) ).
A function is continuous at a left endpoint ( a ) if ( \lim_{x \to a^+} f(x) = f(a) ).
At a right endpoint ( b ), continuity means ( \lim_{x \to b^-} f(x) = f(b) ).
Continuous functions remain continuous at every point in their domain.
The composition of continuous functions remains continuous.

Types of Discontinuity

There are two main types of discontinuities:
- Jump discontinuity, characterized by breaks in the graph.
- Removable discontinuity, occurs where there is a hole in the graph.

Theorems and Concepts

The Intermediate Value Theorem states that if ( y = f(x) ) is continuous on the interval ([a, b]), it takes every value between ( f(a) ) and ( f(b) ).
Average Rate of Change is calculated as the change in distance divided by the change in time.

Specific Examples

For ( f(x) = \sqrt{2x + 3} ):
- Continuous on ([-3/2, \infty)); discontinuous on ((- \infty, -3/2)).
A continuous extension of ( f(x) = \frac{x^2 - 9}{x + 3} ) at ( x = -3 ) is ( g(x) = x - 3 ).

Limits

The limit ( \lim_{x \to 0} \frac{\sin x}{x} = 1 ) and ( \lim_{x \to 0} \frac{\tan x}{x} = 1 ).
Application of Limit Rules:
- Sum Rule: ( \lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) ).
- Difference Rule: ( \lim_{x \to c} (f(x) - g(x)) = \lim_{x \to c} f(x) - \lim_{x \to c} g(x) ).
- Product Rule: ( \lim_{x \to c} (f(x)g(x)) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) ).
- Quotient Rule (denominator cannot be zero): ( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} ).
- Power Rule: ( \lim_{x \to c} f(x)^t = (\lim_{x \to c} f(x))^t ).

Asymptotes and Behavior

Vertical asymptotes are related to the behavior of a function as it approaches specific points, often linked with discontinuities.

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This quiz focuses on key concepts from Chapter 2 of AP Calculus AB, specifically reviewing continuous functions and their properties. Use these flashcards to test your understanding and prepare for your exams. Perfect for strengthening your calculus skills!