Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and studied how it is employed to differentiate a product of two functions. In particular, recall that if ƒ and g are differentiable functions of x, then a. For each of the following functions, use the Product Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of g(x) should be labeled g'(x)). d -[ƒ(x) · g(x)] = f(x) · g'(x) + g(x) · ƒ'(x). dx i. g(x) = x sin(x) iii. p(x) = x ln(x) v. r(x) = e* sin(x) b. Use your work in (a) to help you evaluate the following indefinite integrals. Use differentiation to check your work. İ. ·[re² [2.r cos iii. V. xe te dữ 2x cos(x) — x² sin(x) dx 1 + ln(x) dx ii. h(x) = xe* iv. q(x) = x² cos(x) ii. iv. [e² (sin(x) + cos(x)) da [r cos(x) + sin(x) da
Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and studied how it is employed to differentiate a product of two functions. In particular, recall that if ƒ and g are differentiable functions of x, then a. For each of the following functions, use the Product Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of g(x) should be labeled g'(x)). d -[ƒ(x) · g(x)] = f(x) · g'(x) + g(x) · ƒ'(x). dx i. g(x) = x sin(x) iii. p(x) = x ln(x) v. r(x) = e* sin(x) b. Use your work in (a) to help you evaluate the following indefinite integrals. Use differentiation to check your work. İ. ·[re² [2.r cos iii. V. xe te dữ 2x cos(x) — x² sin(x) dx 1 + ln(x) dx ii. h(x) = xe* iv. q(x) = x² cos(x) ii. iv. [e² (sin(x) + cos(x)) da [r cos(x) + sin(x) da
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 need help with a and b please, thank you!
![Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and
studied how it is employed to differentiate a product of two functions. In
particular, recall that if f and g are differentiable functions of x, then
a. For each of the following functions, use the Product Rule to find the
function's derivative. Be sure to label each derivative by name (e.g., the
derivative of g(x) should be labeled g'(x)).
i.
d
-[ƒf(x) · g(x)] = f(x) · g'(x) + g(x) · ƒ'(x).
dx
i. g(x) = x sin(x)
iii. p(x) = x ln(x)
v. r(x) = e* sin(x)
b. Use your work in (a) to help you evaluate the following indefinite
integrals. Use differentiation to check your work.
iii.
V.
xe te da
[2x
[1+
1 + ln(x) dx
ii. h(x):
=xe*
iv. q(x) = x² cos(x)
2x cos(x) — x² sin(x) dx
ii.
iv.
e* (sin(x) + cos(x)) dx
[x
x cos(x) + sin(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives
you were asked to calculate in (a). Each integrand in (b) is precisely the
result of differentiating one of the products of basic functions found in
(a). To see what happens when an integrand is still a product but not
necessarily the result of differentiating an elementary product, we
consider how to evaluate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce3589f0-5608-4fb5-a11c-ce1805f2c8e6%2Fd20cf7b8-0897-42ac-885e-090c5b97b60e%2Fxrv1h8t_processed.png&w=3840&q=75)
Transcribed Image Text:Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and
studied how it is employed to differentiate a product of two functions. In
particular, recall that if f and g are differentiable functions of x, then
a. For each of the following functions, use the Product Rule to find the
function's derivative. Be sure to label each derivative by name (e.g., the
derivative of g(x) should be labeled g'(x)).
i.
d
-[ƒf(x) · g(x)] = f(x) · g'(x) + g(x) · ƒ'(x).
dx
i. g(x) = x sin(x)
iii. p(x) = x ln(x)
v. r(x) = e* sin(x)
b. Use your work in (a) to help you evaluate the following indefinite
integrals. Use differentiation to check your work.
iii.
V.
xe te da
[2x
[1+
1 + ln(x) dx
ii. h(x):
=xe*
iv. q(x) = x² cos(x)
2x cos(x) — x² sin(x) dx
ii.
iv.
e* (sin(x) + cos(x)) dx
[x
x cos(x) + sin(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives
you were asked to calculate in (a). Each integrand in (b) is precisely the
result of differentiating one of the products of basic functions found in
(a). To see what happens when an integrand is still a product but not
necessarily the result of differentiating an elementary product, we
consider how to evaluate
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning