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c. Based on your experience in parts (a) and (b), conjecture an antiderivative for each of the following functions. Test your conjectures by computing the derivative of each proposed antiderivative. ii. b(x) = (4x + 7)¹¹ i. a(x) = cos(x) iii. c(x) =xe

c. Based on your experience in parts (a) and (b), conjecture an antiderivative for each of the following functions. Test your conjectures by computing the derivative of each proposed antiderivative. ii. b(x) = (4x + 7)¹¹ i. a(x) = cos(x) iii. c(x) =xe

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Only need help with part c, please.

 

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Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if u is a differentiable function of x, and f is a differentiable function of u(x), then −[ƒ(u(x))] = f'(u(x)) · u'(x). น In words, we say that the derivative of a composite function c(x) = f(u(x)), where f is considered the "outer" function and u the “inner" function, is "the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function." a. For each of the following functions, use the Chain 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. g(x) = e³x 3x dx 3x i. m(x) = e³x İİİ. p(x) = arctan(2x) V. r(x) = 34-11x b. For each of the following functions, use your work in (a) to help you determ the general antiderivative of the function. Label each antiderivative by name (e.g., the antiderivative of m should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. ii. h(x) = sin(5x + 1) iv. q(x) = (2 – 7x)ª iii. s(x) = = ii. n(x) = cos(5x + 1) iv. v(x) = (2 — 7x)³ 1 1+4x² V. w(x) = 34-11x c. Based on your experience in parts (a) and (b), conjecture an antiderivative
Transcribed Image Text:Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if u is a differentiable function of x, and f is a differentiable function of u(x), then −[ƒ(u(x))] = f'(u(x)) · u'(x). น In words, we say that the derivative of a composite function c(x) = f(u(x)), where f is considered the "outer" function and u the “inner" function, is "the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function." a. For each of the following functions, use the Chain 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. g(x) = e³x 3x dx 3x i. m(x) = e³x İİİ. p(x) = arctan(2x) V. r(x) = 34-11x b. For each of the following functions, use your work in (a) to help you determ the general antiderivative of the function. Label each antiderivative by name (e.g., the antiderivative of m should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. ii. h(x) = sin(5x + 1) iv. q(x) = (2 – 7x)ª iii. s(x) = = ii. n(x) = cos(5x + 1) iv. v(x) = (2 — 7x)³ 1 1+4x² V. w(x) = 34-11x c. Based on your experience in parts (a) and (b), conjecture an antiderivative
In words, we say that the derivative of a composite function c(x) = f(u(x)), where f is considered the "outer" function and u the "inner" function, is "the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function." a. For each of the following functions, use the Chain 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. g(x) = e³x 3x iii. p(x) = arctan(2x) V. r(x) = 34-11x b. For each of the following functions, use your work in (a) to help you determine the general antiderivative of the function. Label each antiderivative by name (e.g., the antiderivative of m should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. ii. h(x) = sin(5x + 1) iv. g(x) = (2 - 7x) 4 i. m(x) = e³x 1 1+ 4x² V. w(x) = 34-11æ c. Based on your experience in parts (a) and (b), conjecture an antiderivative for each of the following functions. Test your conjectures by computing the derivative of each proposed antiderivative. ii. b(x) = (4x + 7)¹¹ iii. s(x) = i. a(x) = cos(x) iii. c(2) = xe ii. n(x) = cos(5x + 1) iv. v(x) = (2 – 7x)³
Transcribed Image Text:In words, we say that the derivative of a composite function c(x) = f(u(x)), where f is considered the "outer" function and u the "inner" function, is "the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function." a. For each of the following functions, use the Chain 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. g(x) = e³x 3x iii. p(x) = arctan(2x) V. r(x) = 34-11x b. For each of the following functions, use your work in (a) to help you determine the general antiderivative of the function. Label each antiderivative by name (e.g., the antiderivative of m should be called M). In addition, check your work by computing the derivative of each proposed antiderivative. ii. h(x) = sin(5x + 1) iv. g(x) = (2 - 7x) 4 i. m(x) = e³x 1 1+ 4x² V. w(x) = 34-11æ c. Based on your experience in parts (a) and (b), conjecture an antiderivative for each of the following functions. Test your conjectures by computing the derivative of each proposed antiderivative. ii. b(x) = (4x + 7)¹¹ iii. s(x) = i. a(x) = cos(x) iii. c(2) = xe ii. n(x) = cos(5x + 1) iv. v(x) = (2 – 7x)³
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