AL THEOREM OF CALCULUS I ,b] then the function g(x) = f(x)dx 1, differentiable on (a,b) and g'(x) = f(x). AL THEOREM OF CALCULUS II ,b] and Fis any antiderivative of f, then f(x)dx = F(b) – F(a).
AL THEOREM OF CALCULUS I ,b] then the function g(x) = f(x)dx 1, differentiable on (a,b) and g'(x) = f(x). AL THEOREM OF CALCULUS II ,b] and Fis any antiderivative of f, then f(x)dx = F(b) – F(a).
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...
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Question
Only answer a through h, and only answer using the formulas given. Thank you
(P1)
![FUNDAMENTAL THEOREM OF CALCULUS I
If fis continuous on [a,b] then the function
BASIC ANTIDERIVATIVES
Constant
= ax +C, where a ER
%3D
Power:
is continuous on [a, b], dfferentiable on (a, b) and gʻ'(x) = f(x).
(rds =
+C, where r + - 1
r +1
FUNDAMENTAL THEOREM OF CALCULUS II
If fis continuous on [a,b] and F is any antiderivative of f, then
= In |x|+C
Exponential:
f(x)dx = F(b) – F(a).
b
+C, where b E (0,00)
In(b)
Trigonometric:
NET CHANGE THEOREM
cos(x)dx = sin(x) +C
If F' is continuous on [a, b), then
Jamer-
sec-(endx = tan(x) + C
= - cos(x) +C
F(x)dx = F(b) – F(a).
VARIABLE SUBSTITUTION
sec(x)tan(x)d x = sec(x) + C
If u = g(x) is differentiable whose range contains [a, b] and fis
continuous on [a, b), then
Joscondx = - cot(2) + C
r)cot(x)dx = - csc(x) +C
VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS
tan(x)dx = - In|cos(x)|+C_
If u = g(x) is differentiable whose range contains [a, b] and fis
continuous on [a, b), then
Jcot(o)d x = In |sin(<)| + c
f(u)du.
sec(x)dx = In|sec(x) + tan(x)| + C
gla)
x)dx = In[cse(x) – cot(x)| +C
VOLUMES OF SOLIDS OF REVOLUTION
Inverse Trigonometric:
Revolving about x Revolving about y-
xp
= arcsin(x) + C, where x *±1
-ахis
axis
Disks/Washers Integrate x variable Integrate y variable
dx
= arctan(x) + C
1+x2
Cylindrical Shells Integrate y variable Integrate x variable
Нурerbolic
AVERAGE VALUE OF A FUNCTION
cosh(x)d x = sinh(x) +C
Iffis continuous on [a, b], then
sinh(x) +C = cosh(x) + C
Savg =
f(x)dx.
b- a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73b00b2a-4fa8-4b66-93aa-a27c1d9b51ff%2F75d9ae0f-7929-45ac-91d0-7b14168ac6e0%2Fnq5mfrj_processed.png&w=3840&q=75)
Transcribed Image Text:FUNDAMENTAL THEOREM OF CALCULUS I
If fis continuous on [a,b] then the function
BASIC ANTIDERIVATIVES
Constant
= ax +C, where a ER
%3D
Power:
is continuous on [a, b], dfferentiable on (a, b) and gʻ'(x) = f(x).
(rds =
+C, where r + - 1
r +1
FUNDAMENTAL THEOREM OF CALCULUS II
If fis continuous on [a,b] and F is any antiderivative of f, then
= In |x|+C
Exponential:
f(x)dx = F(b) – F(a).
b
+C, where b E (0,00)
In(b)
Trigonometric:
NET CHANGE THEOREM
cos(x)dx = sin(x) +C
If F' is continuous on [a, b), then
Jamer-
sec-(endx = tan(x) + C
= - cos(x) +C
F(x)dx = F(b) – F(a).
VARIABLE SUBSTITUTION
sec(x)tan(x)d x = sec(x) + C
If u = g(x) is differentiable whose range contains [a, b] and fis
continuous on [a, b), then
Joscondx = - cot(2) + C
r)cot(x)dx = - csc(x) +C
VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS
tan(x)dx = - In|cos(x)|+C_
If u = g(x) is differentiable whose range contains [a, b] and fis
continuous on [a, b), then
Jcot(o)d x = In |sin(<)| + c
f(u)du.
sec(x)dx = In|sec(x) + tan(x)| + C
gla)
x)dx = In[cse(x) – cot(x)| +C
VOLUMES OF SOLIDS OF REVOLUTION
Inverse Trigonometric:
Revolving about x Revolving about y-
xp
= arcsin(x) + C, where x *±1
-ахis
axis
Disks/Washers Integrate x variable Integrate y variable
dx
= arctan(x) + C
1+x2
Cylindrical Shells Integrate y variable Integrate x variable
Нурerbolic
AVERAGE VALUE OF A FUNCTION
cosh(x)d x = sinh(x) +C
Iffis continuous on [a, b], then
sinh(x) +C = cosh(x) + C
Savg =
f(x)dx.
b- a
![1. Determine the following using the graph off provided below. If the value is infinite, specify oo or - o. If an answer
does not exist, explain why.
lim
1-1 2f(x)
d. lim f(x)
lim f(x)
b. lim f(x)
a.
с.
h. Average value off
on [-4, – 1]
f(x)dx
f(x)dx
lim
dx
dx
g.
5
i. List and classify critical points of f as relative maxima, minima, or neither.
j. Find the volume of the solid formed by revolving the region bounded by f, x = – 7,x = - 4 and y = 0 about
the y-axis Shade the region on the graph](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73b00b2a-4fa8-4b66-93aa-a27c1d9b51ff%2F75d9ae0f-7929-45ac-91d0-7b14168ac6e0%2F6oiymm9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Determine the following using the graph off provided below. If the value is infinite, specify oo or - o. If an answer
does not exist, explain why.
lim
1-1 2f(x)
d. lim f(x)
lim f(x)
b. lim f(x)
a.
с.
h. Average value off
on [-4, – 1]
f(x)dx
f(x)dx
lim
dx
dx
g.
5
i. List and classify critical points of f as relative maxima, minima, or neither.
j. Find the volume of the solid formed by revolving the region bounded by f, x = – 7,x = - 4 and y = 0 about
the y-axis Shade the region on the graph
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