AP Calculus AB Units 1 to 5 Review Study Guide and Practice Questions
Unit 1 Unit 1
If the right-hand limit does not equal the left-hand limit, The limit is DNE
then...
If the graph of a function shows a hole or open-circle point, the function does not exist there
then...
What should you do when asked to find the limit of a Find the limit as x approaches A of g(x). Let that limit value equal B.
composite function? Then, find the limit as x approaches B of f(x)
ex) limit as x approaches a of f(g(x))
What should you do when asked to find the limit of a Find both the left and right hand limits of both functions
compound function? ex) you will need the left-hand limits of f(x) and g(x), then add them together to find
ex) limit as x approaches 1 of [ f(x) + g(x) ] the left compound limit
THEN, find the right-hand limits of f(x) and g(x) and add them together to find the
right compound limit.
(IF the right compound limit does not equal the left compound limit, the limit DNE.)
What are the rules for finding limits at infinity of rational Look at the degree of the numerator and degree of the denominator. (Let the
functions? degree of the numerator = N and the degree of the denominator = D)
- If N = D, the limit is a ratio of the leading coefficients
- If N < D (or D > N), the limit is zero
- If N > D, the limit DNE
What are the rules for determining limits at infinity of non- Look at the degree and the sign of the leading coefficient to determine end
rational functions? behavior!
- For even positive end behavior, the limit as x approaches both infinities is positive
infinity.
- For even negative end behavior, the limit as x approaches both infinities is
negative infinity
- For odd positive end behavior, the limit as x approaches negative infinity is
negative infinity. The limit as x approaches positive infinity is positive infinity.
- For odd negative end behavior, the limit as x approaches negative infinity is
positive infinity. The limit as x approaches positive infinity is negative infinity.
If the numerator or denominator has a radical, what should Remember that square roots can be positive or negative!
you do? ex) square root of 4 can be 2 or -2
Determine whether to use the positive or negative root by which infinity x is
approaching:
- if x approaches positive infinity, use the positive root
- if x approaches negative infinity, use the negative root
What do you do to find the limits at a horizontal Use the same rules as limits at infinity, because you will be taking the limit as x
asymptote? approaches both positive and negative infinity.
(If there is a radical, remember to include the positive and negative root, because
you are approaching both infinities)
What are the rules for absolute value functions as Recall that the absolute value of x is a piecewise function
piecewise functions? - If x > 0, then f(x) = x. If x < 0, then f(x) = -x
|x| / x is also piecewise:
- If x > 0, then |x| / x = 1.
- If x < 0, then |x| / x = -1.
Therefore, the limit of an absolute value function will always be 1 or -1, unless being
multiplied/divided by a factor.
Grade A+ | Updated
, AP Calculus AB Units 1 to 5 Review Study Guide and Practice Questions
What are the steps to finding the limit of an absolute value 1) Factor out any numbers being multiplied or divided
function? 2) If you aren't being asked for a one-sided limit, separate the limit into left- and
right-hand limits and solve
3) For a left-hand limit, plug in a number slightly less than what x is approaching.
For a right-hand limit, plug in a number slightly greater than what x is approaching.
4) Multiply by the factor (if applicable).
5) Ensure that the left hand limit = the right hand limit; otherwise, the limit DNE
What is the special trig limit? The limit as x approaches zero of sin(c0) / (d0)
The limit will evaluate to c/d.
What are indeterminate forms of limits? a limit whose value is a fraction over zero
What should you do when the limit evaluates to 0/0 ? "More work"!
- Factor the numerator and denominator
- Find a common factor that can be cancelled out
- Use direct substitution with the remaining factors: plug in the number that x is
approaching, and solve
OR use L'Hospital's Rule
What should you do when the limit evaluates to #/0 ? Vertical asymptote!
We can still determine if f(x) approaches positive or negative infinity:
- Plug in a number very close to the left and right of the value that x is approaching.
(ex: if x approaches 8, use 7.9 and 8.1)
- If you get +/+ or -/-, then f(x) approaches positive infinity
- If you get -/+ or +/-, then f(x) approaches negative infinity
What should you be sure to do on an FRQ when the limit You can draw an arrow to 0/0 (instead of an equal sign) or write "indeterminate form
evaluates to indeterminate form? 0/0".
For a rational function, separately establish that the limit of the top function and the
limit of the bottom function would equal zero if you used direct substitution, and that
this yields (arrow) the indeterminate form 0/0.
DO NOT write that the limit = 0/0
What is the 3-part definition of continuity? 1) the limit of f(x) as x approaches c must exist
(i.e., the left limit must equal the right limit)
2) f(c) must exist
3) the limit of f(x) as x approaches c must equal f(c)
What are the 3 types of discontinuities, and where are they Jump discontinuity, removable point discontinuity, and vertical/horizontal
found? asymptotes.
- Jump discontinuities occur when the left and right limits are not equal
- Removable point discontinuities occur when the function value does not equal the
limit value
- vertical asymptotes occur when the limit approaches positive or negative infinity
- horizontal asymptotes occur when the limit as x approaches infinity equals a
number (instead of an infinity, which would indicate end behavior)
What does the intermediate value theorem (IVT) IVT guarantees the existence of a value "c" on the interval [a, b] for which f(c) = d.
guarantee, and what are the conditions? Conditions:
- f(x) must be continuous on the given interval [a,b]
- c must be between a and b
- f(c) must be between f(a) and f(b)
When solving IVT problems, make sure to establish continuity first!!
Unit 2 Unit 2
What does the derivative represent? The slope of the tangent line OR the rate of change at a particular value of x
Grade A+ | Updated
Unit 1 Unit 1
If the right-hand limit does not equal the left-hand limit, The limit is DNE
then...
If the graph of a function shows a hole or open-circle point, the function does not exist there
then...
What should you do when asked to find the limit of a Find the limit as x approaches A of g(x). Let that limit value equal B.
composite function? Then, find the limit as x approaches B of f(x)
ex) limit as x approaches a of f(g(x))
What should you do when asked to find the limit of a Find both the left and right hand limits of both functions
compound function? ex) you will need the left-hand limits of f(x) and g(x), then add them together to find
ex) limit as x approaches 1 of [ f(x) + g(x) ] the left compound limit
THEN, find the right-hand limits of f(x) and g(x) and add them together to find the
right compound limit.
(IF the right compound limit does not equal the left compound limit, the limit DNE.)
What are the rules for finding limits at infinity of rational Look at the degree of the numerator and degree of the denominator. (Let the
functions? degree of the numerator = N and the degree of the denominator = D)
- If N = D, the limit is a ratio of the leading coefficients
- If N < D (or D > N), the limit is zero
- If N > D, the limit DNE
What are the rules for determining limits at infinity of non- Look at the degree and the sign of the leading coefficient to determine end
rational functions? behavior!
- For even positive end behavior, the limit as x approaches both infinities is positive
infinity.
- For even negative end behavior, the limit as x approaches both infinities is
negative infinity
- For odd positive end behavior, the limit as x approaches negative infinity is
negative infinity. The limit as x approaches positive infinity is positive infinity.
- For odd negative end behavior, the limit as x approaches negative infinity is
positive infinity. The limit as x approaches positive infinity is negative infinity.
If the numerator or denominator has a radical, what should Remember that square roots can be positive or negative!
you do? ex) square root of 4 can be 2 or -2
Determine whether to use the positive or negative root by which infinity x is
approaching:
- if x approaches positive infinity, use the positive root
- if x approaches negative infinity, use the negative root
What do you do to find the limits at a horizontal Use the same rules as limits at infinity, because you will be taking the limit as x
asymptote? approaches both positive and negative infinity.
(If there is a radical, remember to include the positive and negative root, because
you are approaching both infinities)
What are the rules for absolute value functions as Recall that the absolute value of x is a piecewise function
piecewise functions? - If x > 0, then f(x) = x. If x < 0, then f(x) = -x
|x| / x is also piecewise:
- If x > 0, then |x| / x = 1.
- If x < 0, then |x| / x = -1.
Therefore, the limit of an absolute value function will always be 1 or -1, unless being
multiplied/divided by a factor.
Grade A+ | Updated
, AP Calculus AB Units 1 to 5 Review Study Guide and Practice Questions
What are the steps to finding the limit of an absolute value 1) Factor out any numbers being multiplied or divided
function? 2) If you aren't being asked for a one-sided limit, separate the limit into left- and
right-hand limits and solve
3) For a left-hand limit, plug in a number slightly less than what x is approaching.
For a right-hand limit, plug in a number slightly greater than what x is approaching.
4) Multiply by the factor (if applicable).
5) Ensure that the left hand limit = the right hand limit; otherwise, the limit DNE
What is the special trig limit? The limit as x approaches zero of sin(c0) / (d0)
The limit will evaluate to c/d.
What are indeterminate forms of limits? a limit whose value is a fraction over zero
What should you do when the limit evaluates to 0/0 ? "More work"!
- Factor the numerator and denominator
- Find a common factor that can be cancelled out
- Use direct substitution with the remaining factors: plug in the number that x is
approaching, and solve
OR use L'Hospital's Rule
What should you do when the limit evaluates to #/0 ? Vertical asymptote!
We can still determine if f(x) approaches positive or negative infinity:
- Plug in a number very close to the left and right of the value that x is approaching.
(ex: if x approaches 8, use 7.9 and 8.1)
- If you get +/+ or -/-, then f(x) approaches positive infinity
- If you get -/+ or +/-, then f(x) approaches negative infinity
What should you be sure to do on an FRQ when the limit You can draw an arrow to 0/0 (instead of an equal sign) or write "indeterminate form
evaluates to indeterminate form? 0/0".
For a rational function, separately establish that the limit of the top function and the
limit of the bottom function would equal zero if you used direct substitution, and that
this yields (arrow) the indeterminate form 0/0.
DO NOT write that the limit = 0/0
What is the 3-part definition of continuity? 1) the limit of f(x) as x approaches c must exist
(i.e., the left limit must equal the right limit)
2) f(c) must exist
3) the limit of f(x) as x approaches c must equal f(c)
What are the 3 types of discontinuities, and where are they Jump discontinuity, removable point discontinuity, and vertical/horizontal
found? asymptotes.
- Jump discontinuities occur when the left and right limits are not equal
- Removable point discontinuities occur when the function value does not equal the
limit value
- vertical asymptotes occur when the limit approaches positive or negative infinity
- horizontal asymptotes occur when the limit as x approaches infinity equals a
number (instead of an infinity, which would indicate end behavior)
What does the intermediate value theorem (IVT) IVT guarantees the existence of a value "c" on the interval [a, b] for which f(c) = d.
guarantee, and what are the conditions? Conditions:
- f(x) must be continuous on the given interval [a,b]
- c must be between a and b
- f(c) must be between f(a) and f(b)
When solving IVT problems, make sure to establish continuity first!!
Unit 2 Unit 2
What does the derivative represent? The slope of the tangent line OR the rate of change at a particular value of x
Grade A+ | Updated
