AB Calculus Theorems Study Guide for AP Calculus Exam Page 1
1. The Intermediate Value Theorem:
If a function f is continuous over the closed interval [a, b],
and k is some number between f(a) and f(b), then the IVT guarantees
that there is at least one number c between a and b such that f(c) = k.
iiiiiiiii
A function continuous on [a, b] must take on each value from f(a) to f(b).
2. The Extreme Value Theorem:
If a function f is continuous over the closed interval [a, b],
then the EVT guarantees that f attains an absolute maximum value
and an absolute minimum value on [a, b].
A function continuous on [a, b] must have global extreme values.
3. The Candidates Theorem: Absolute (global) extrema of a function on a closed interval can
only occur at the endpoints or at critical points where the first derivative is zero or fails to exist.
4. The Mean Value Theorem: (Average Rate of Change or Average Slope)
i.li
If a function f is continuous over [a, b] and differentiable inside (a, b),
f(b) - f(a)
then the MVT guarantees a number c inside (a, b) where f ¢(c) = .
numberb etween
b - a
some
a and bha athe averageslope
itsthe average rate of change.
between and b as slope
MVT: There is at least one point where the instantaneous rate equals
5. The Fundamental Theorems of Calculus (FTC):
x
Part 1: If g(x) = ò a
f(t) dt , then g¢(x) = f(x) when g is an integral or accumulation function of f.
So g(x), the area under f, is changing at an instantaneous rate equal to the y-value of f at x.
d é x ù = f(x). By the chain rule, d é g(x) f(t) dt ù = f(g(x)) × g¢(x).
dx êë ò a dx êë ò a
In symbols, f(t) dt
úû úû
b
Part 2: ò a
f(x) dx = F(b) – F(a) where F is any antiderivative of f on the interval [a, b].
b
Since f(x) = F¢(x), we can rewrite this as ò a
F¢(x) dx = F(b) – F(a).
Application: We can find the net change in a quantity given its rate of change. The definite
integral of a rate of change of a quantity is equal to the total amount of change in that quantity.
6. Inverse Function Derivative Theorem: If g is the inverse of f such that g(a) = b and f(b) = a,
1 1 1
then g¢(a) = . In general, g¢(x) = or where y = g(x).
f ¢(b) f ¢(g(x)) f ¢(y)
, AB Calculus Definitions Study Guide for AP Calculus Exam Page 2
7. Definition of Continuity at a point: A function f is continuous at x = c provided that
f(c) exists, lim f(x) exists, and lim f(x) = f(c). Both one-sided limits must equal f(c).
%→ ' %→ '
8. Asymptotes: A line that the graph of a function approaches as x or y increases to infinity.
a. A vertical asymptote exists at x = a if the limit of f as x approaches a is infinite (± ¥).
b. A horizontal asymptote exists at y = b if the limit of f as x approaches infinity is b.
9. Discontinuities of a Function: A function is discontinuous at x = c when there is:
a. a point discontinuity (hole in graph) which is a removable discontinuity.
b. an infinite discontinuity (vertical asymptote) which is a nonremovable discontinuity.
c. a jump discontinuity (one-sided limits differ) which is a nonremovable discontinuity.
Note: A function is continuous on an interval if it is continuous at each point within the interval.
The graph of a function has no breaks if it is continuous on that interval.
10. Definition: The derivative at the point x = a is the limit of the slope of f approaching x = a.
- % . -(,)
a. The slope of the graph of f at x = a is given by f ¢(a) = lim .
%→, %.,
- , 1/ .-(,)
b. The instantaneous rate of change of f at x = a is given by f ¢(a) = lim .
/→0 /
where h is a small change in x.
- %1/ .-(%)
11. Definition: The derivative of function f is f ¢(x) = lim which tells us the
/→0 /
23
instantaneous rate of change of f at any point where this limit exists. So f ¢(x) = .
2%
12. Definition: A function is differentiable at x = c if f ¢(c) exists as a finite number.
A function is not differentiable at x = c if there is a discontinuity, a sharp turn, or a vertical
tangent line at x = c. The graph of f must be smooth and continuous to be differentiable.
13. The First Derivative Local Extrema Test: Suppose that f ¢(c) = 0 or fails to exist.
a. If f ¢ changes from positive to negative at c, then f has a local maximum at c.
b. If f ¢ changes from negative to positive at c, then f has a local minimum at c.
c. If f ¢ does not change signs at c, then f has no local minimum nor maximum at c.
14. The Second Derivative Local Extrema Test: Suppose that f ¢(c) = 0 or fails to exist.
a. If f ¢¢(c) > 0, then f has a local minimum at c (since f is concave up at that point).
b. If f ¢¢(c) < 0, then f has a local maximum at c (since f is concave down at that point).
c. If f ¢¢(c) = 0, then this test fails so you need to use the first derivative test above.
15. The local linear approximation for a function f at point x = a is y = f(a) + f ¢(a) (x – a).
The local linear approximation of f is the equation of the line tangent to the graph at point a.
1. The Intermediate Value Theorem:
If a function f is continuous over the closed interval [a, b],
and k is some number between f(a) and f(b), then the IVT guarantees
that there is at least one number c between a and b such that f(c) = k.
iiiiiiiii
A function continuous on [a, b] must take on each value from f(a) to f(b).
2. The Extreme Value Theorem:
If a function f is continuous over the closed interval [a, b],
then the EVT guarantees that f attains an absolute maximum value
and an absolute minimum value on [a, b].
A function continuous on [a, b] must have global extreme values.
3. The Candidates Theorem: Absolute (global) extrema of a function on a closed interval can
only occur at the endpoints or at critical points where the first derivative is zero or fails to exist.
4. The Mean Value Theorem: (Average Rate of Change or Average Slope)
i.li
If a function f is continuous over [a, b] and differentiable inside (a, b),
f(b) - f(a)
then the MVT guarantees a number c inside (a, b) where f ¢(c) = .
numberb etween
b - a
some
a and bha athe averageslope
itsthe average rate of change.
between and b as slope
MVT: There is at least one point where the instantaneous rate equals
5. The Fundamental Theorems of Calculus (FTC):
x
Part 1: If g(x) = ò a
f(t) dt , then g¢(x) = f(x) when g is an integral or accumulation function of f.
So g(x), the area under f, is changing at an instantaneous rate equal to the y-value of f at x.
d é x ù = f(x). By the chain rule, d é g(x) f(t) dt ù = f(g(x)) × g¢(x).
dx êë ò a dx êë ò a
In symbols, f(t) dt
úû úû
b
Part 2: ò a
f(x) dx = F(b) – F(a) where F is any antiderivative of f on the interval [a, b].
b
Since f(x) = F¢(x), we can rewrite this as ò a
F¢(x) dx = F(b) – F(a).
Application: We can find the net change in a quantity given its rate of change. The definite
integral of a rate of change of a quantity is equal to the total amount of change in that quantity.
6. Inverse Function Derivative Theorem: If g is the inverse of f such that g(a) = b and f(b) = a,
1 1 1
then g¢(a) = . In general, g¢(x) = or where y = g(x).
f ¢(b) f ¢(g(x)) f ¢(y)
, AB Calculus Definitions Study Guide for AP Calculus Exam Page 2
7. Definition of Continuity at a point: A function f is continuous at x = c provided that
f(c) exists, lim f(x) exists, and lim f(x) = f(c). Both one-sided limits must equal f(c).
%→ ' %→ '
8. Asymptotes: A line that the graph of a function approaches as x or y increases to infinity.
a. A vertical asymptote exists at x = a if the limit of f as x approaches a is infinite (± ¥).
b. A horizontal asymptote exists at y = b if the limit of f as x approaches infinity is b.
9. Discontinuities of a Function: A function is discontinuous at x = c when there is:
a. a point discontinuity (hole in graph) which is a removable discontinuity.
b. an infinite discontinuity (vertical asymptote) which is a nonremovable discontinuity.
c. a jump discontinuity (one-sided limits differ) which is a nonremovable discontinuity.
Note: A function is continuous on an interval if it is continuous at each point within the interval.
The graph of a function has no breaks if it is continuous on that interval.
10. Definition: The derivative at the point x = a is the limit of the slope of f approaching x = a.
- % . -(,)
a. The slope of the graph of f at x = a is given by f ¢(a) = lim .
%→, %.,
- , 1/ .-(,)
b. The instantaneous rate of change of f at x = a is given by f ¢(a) = lim .
/→0 /
where h is a small change in x.
- %1/ .-(%)
11. Definition: The derivative of function f is f ¢(x) = lim which tells us the
/→0 /
23
instantaneous rate of change of f at any point where this limit exists. So f ¢(x) = .
2%
12. Definition: A function is differentiable at x = c if f ¢(c) exists as a finite number.
A function is not differentiable at x = c if there is a discontinuity, a sharp turn, or a vertical
tangent line at x = c. The graph of f must be smooth and continuous to be differentiable.
13. The First Derivative Local Extrema Test: Suppose that f ¢(c) = 0 or fails to exist.
a. If f ¢ changes from positive to negative at c, then f has a local maximum at c.
b. If f ¢ changes from negative to positive at c, then f has a local minimum at c.
c. If f ¢ does not change signs at c, then f has no local minimum nor maximum at c.
14. The Second Derivative Local Extrema Test: Suppose that f ¢(c) = 0 or fails to exist.
a. If f ¢¢(c) > 0, then f has a local minimum at c (since f is concave up at that point).
b. If f ¢¢(c) < 0, then f has a local maximum at c (since f is concave down at that point).
c. If f ¢¢(c) = 0, then this test fails so you need to use the first derivative test above.
15. The local linear approximation for a function f at point x = a is y = f(a) + f ¢(a) (x – a).
The local linear approximation of f is the equation of the line tangent to the graph at point a.
