a critical point of fif either f'(c) = 0 orf'(c) does not exist. ) is an absolute maximum value of fiff(c) 2 f(x) for all x in the domain of f. ) is an absolute minimum value of fiff(c) < f(x) for all x in the domain of f. ) is a relative minimum value of fiff(c) 2 f(x) for all x near c. ) is a relative maximum value of f if f(c) < f(x) for all x near c. EXTREME VALUE THEOREM FERMAT'S THEOREM fis continuous on a closed and bounded interval [a,b], then f attains a maximum and minimum on the interval. Iff(c) is a relative maximum or minimum value, then f'(c) = 0 ROLLE'S THEOREM MEAN VALUE THEOREM f be a function which satisfies the following: Let f be a function which satisfies the following: is continuous on the closed interval [a, b]. Fis differentiable on the open interval (a, b). 1. fis continuous on the closed interval [a, b]. 2. fis differentiable on the open interval (a,b). Then there is some c in the interval (a, b) where (a) = f (b). en there is some c in the interval (a , b) where f'(c) = 0. f(a) – f(b) f'(c) =' a - b CALCULUS AND GRAPHS "(x) > 0 on some interval (a, b), then fis increasing on (a,b). "(x) < 0 on some interval (a, b), then f is decreasing on (a, b). st Derivative Test: Suppose c is a critical point of some functionf. a. Iff'changes from positive to negative at c, then f(c) is a relative maximum. b. Iff'changes from negative to positive at c, then f(c) is a relative minimum. c. If the sign of f' does not change while passing c, then f(c) is neither a maximum nor a minimum (it is a saddle point). "(x) > O on some interval (a, b), then f is concave up on (a, b). "(x) < 0 on some interval (a,b), then f is concave down on (a, b). ond Derivative Test: Suppose c is a critical point of some functionf. a. Iff"(c) < 0, then f(c) is a relative maximum. b. If"(c) > 0, then f(c) is a relative minimum. c. If"(c) = 0, then the test is inconclusive. (Could be a maximum, minimum, or saddle). L'HOPTIAL’S RULE f(x) AREA APPROXIMATION The area under a curve can be approximated using rectangles with
a critical point of fif either f'(c) = 0 orf'(c) does not exist. ) is an absolute maximum value of fiff(c) 2 f(x) for all x in the domain of f. ) is an absolute minimum value of fiff(c) < f(x) for all x in the domain of f. ) is a relative minimum value of fiff(c) 2 f(x) for all x near c. ) is a relative maximum value of f if f(c) < f(x) for all x near c. EXTREME VALUE THEOREM FERMAT'S THEOREM fis continuous on a closed and bounded interval [a,b], then f attains a maximum and minimum on the interval. Iff(c) is a relative maximum or minimum value, then f'(c) = 0 ROLLE'S THEOREM MEAN VALUE THEOREM f be a function which satisfies the following: Let f be a function which satisfies the following: is continuous on the closed interval [a, b]. Fis differentiable on the open interval (a, b). 1. fis continuous on the closed interval [a, b]. 2. fis differentiable on the open interval (a,b). Then there is some c in the interval (a, b) where (a) = f (b). en there is some c in the interval (a , b) where f'(c) = 0. f(a) – f(b) f'(c) =' a - b CALCULUS AND GRAPHS "(x) > 0 on some interval (a, b), then fis increasing on (a,b). "(x) < 0 on some interval (a, b), then f is decreasing on (a, b). st Derivative Test: Suppose c is a critical point of some functionf. a. Iff'changes from positive to negative at c, then f(c) is a relative maximum. b. Iff'changes from negative to positive at c, then f(c) is a relative minimum. c. If the sign of f' does not change while passing c, then f(c) is neither a maximum nor a minimum (it is a saddle point). "(x) > O on some interval (a, b), then f is concave up on (a, b). "(x) < 0 on some interval (a,b), then f is concave down on (a, b). ond Derivative Test: Suppose c is a critical point of some functionf. a. Iff"(c) < 0, then f(c) is a relative maximum. b. If"(c) > 0, then f(c) is a relative minimum. c. If"(c) = 0, then the test is inconclusive. (Could be a maximum, minimum, or saddle). L'HOPTIAL’S RULE f(x) AREA APPROXIMATION The area under a curve can be approximated using rectangles with
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
ONLY SOLVE "e." and "f." And only solve using the given formula sheet. Thanks
(P3.2)
![EXTREMA AND CRITICAL POINTS
c is a critical point of fif either f'(c) = 0 or f'(c) does not exist.
f(c) is an absolute maximum value of fiff (c) >f(x) for all x in the domain of f.
f(c) is an absolute minimum value of fif f(c) < f(x) for all x in the domain of f.
f(c) is a relative minimum value of fiff(c) 2 f (x) for all x near c.
f(c) is a relative maximum value of fiff(c) < f(x) for all x near c.
EXTREME VALUE THEOREM
FERMAT’S THEOREM
If fis continuous on a closed and bounded interval [a, b], then f
Iff(c) is a relative maximum or minimum value, then f'(c) = 0
attains a maximum and minimum on the interval.
ROLLE'S THEOREM
MEAN VALUE THEOREM
Let f be a function which satisfies the following:
Let f be a function which satisfies the following:
1. fis continuous on the closed interval [a, b].
1. fis continuous on the closed interval [a, b].
2. fis differentiable on the open interval (a, b).
2. fis differentiable on the open interval (a, b).
3. f(a) = f (b).
Then there is some c in the interval (a, b) where
Then there is some c in the interval (a, b) where f'(c) = 0.
f(a) – f(b)
f'(c) =
а —Ь
CALCULUS AND GRAPHS
If f'(x) > 0 on some interval (a,b), then f is increasing on (a, b).
If f'(x) < 0 on some interval (a,b), then f is decreasing on (a, b).
First Derivative Test:
Suppose c is a critical point of some function f.
a. Iff' changes from positive to negative at c, then f(c) is a relative maximum.
b. Iff' changes from negative to positive at c, then f(c) is a relative minimum.
c. If the sign of f' does not change while passing c, then f(c) is neither a maximum nor a minimum (it is a saddle point).
If f"(x) > 0 on some interval (a,b), then fis concave up on (a, b).
If f"(x) <0 on some interval (a, b), then f is concave down on (a, b).
Second Derivative Test:
Suppose c is a critical point of some function f.
a. If f"(c) < 0, then f(c) is a relative maximum.
b. If f"(c) > 0, then f(c) is a relative minimum.
c. If f"(c) = 0, then the test is inconclusive. (Could be a maximum, minimum, or saddle).
L’HOPTIAL’S RULE
AREA APPROXIMATION
ƒ(x)
takes either the
The area under a curve can be approximated using rectangles with
either right or left endpoints. You can subdivide an interval [a, b]
Iff and g are differentiable near a and lim =
x-a g(x)
00
or
then
b - a
for n rectangles by taking segments of length -
indeterminate form
00
f(x)
lim
f'(x)
= lim
x-a g'(x)
f(x)dx = lim
f(x*)Ax.
X-a g(x)
n00
i=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3930b8f2-52cf-42d9-bbea-fa5b5c673abd%2F39857417-27f2-4ca4-8405-d6712081a2aa%2Fasxxzx9_processed.png&w=3840&q=75)
Transcribed Image Text:EXTREMA AND CRITICAL POINTS
c is a critical point of fif either f'(c) = 0 or f'(c) does not exist.
f(c) is an absolute maximum value of fiff (c) >f(x) for all x in the domain of f.
f(c) is an absolute minimum value of fif f(c) < f(x) for all x in the domain of f.
f(c) is a relative minimum value of fiff(c) 2 f (x) for all x near c.
f(c) is a relative maximum value of fiff(c) < f(x) for all x near c.
EXTREME VALUE THEOREM
FERMAT’S THEOREM
If fis continuous on a closed and bounded interval [a, b], then f
Iff(c) is a relative maximum or minimum value, then f'(c) = 0
attains a maximum and minimum on the interval.
ROLLE'S THEOREM
MEAN VALUE THEOREM
Let f be a function which satisfies the following:
Let f be a function which satisfies the following:
1. fis continuous on the closed interval [a, b].
1. fis continuous on the closed interval [a, b].
2. fis differentiable on the open interval (a, b).
2. fis differentiable on the open interval (a, b).
3. f(a) = f (b).
Then there is some c in the interval (a, b) where
Then there is some c in the interval (a, b) where f'(c) = 0.
f(a) – f(b)
f'(c) =
а —Ь
CALCULUS AND GRAPHS
If f'(x) > 0 on some interval (a,b), then f is increasing on (a, b).
If f'(x) < 0 on some interval (a,b), then f is decreasing on (a, b).
First Derivative Test:
Suppose c is a critical point of some function f.
a. Iff' changes from positive to negative at c, then f(c) is a relative maximum.
b. Iff' changes from negative to positive at c, then f(c) is a relative minimum.
c. If the sign of f' does not change while passing c, then f(c) is neither a maximum nor a minimum (it is a saddle point).
If f"(x) > 0 on some interval (a,b), then fis concave up on (a, b).
If f"(x) <0 on some interval (a, b), then f is concave down on (a, b).
Second Derivative Test:
Suppose c is a critical point of some function f.
a. If f"(c) < 0, then f(c) is a relative maximum.
b. If f"(c) > 0, then f(c) is a relative minimum.
c. If f"(c) = 0, then the test is inconclusive. (Could be a maximum, minimum, or saddle).
L’HOPTIAL’S RULE
AREA APPROXIMATION
ƒ(x)
takes either the
The area under a curve can be approximated using rectangles with
either right or left endpoints. You can subdivide an interval [a, b]
Iff and g are differentiable near a and lim =
x-a g(x)
00
or
then
b - a
for n rectangles by taking segments of length -
indeterminate form
00
f(x)
lim
f'(x)
= lim
x-a g'(x)
f(x)dx = lim
f(x*)Ax.
X-a g(x)
n00
i=1
![3. Below is the graph of y = f'(x), which is the derivative for some function f. Using the graph of the derivative of f,
determine the following. If any cannot be determined based on the graph alone, explain why.
y =f'(x)
BE CAREFUL WHEN USING [] OR ( ) IN INTERVAL NOTATION.
a. The number of critical points of f
b. The intervals on which f is increasing
5
5
c. The intervals on which f is concave up
d. The x values where f has a local minimum
5
e. f(3)
f. The number of critical points of f'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3930b8f2-52cf-42d9-bbea-fa5b5c673abd%2F39857417-27f2-4ca4-8405-d6712081a2aa%2F2hkxftb_processed.png&w=3840&q=75)
Transcribed Image Text:3. Below is the graph of y = f'(x), which is the derivative for some function f. Using the graph of the derivative of f,
determine the following. If any cannot be determined based on the graph alone, explain why.
y =f'(x)
BE CAREFUL WHEN USING [] OR ( ) IN INTERVAL NOTATION.
a. The number of critical points of f
b. The intervals on which f is increasing
5
5
c. The intervals on which f is concave up
d. The x values where f has a local minimum
5
e. f(3)
f. The number of critical points of f'
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