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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.R.105
Chapter 4, Problem 4.R.105

104–107. Functions from derivatives Find the function f with the following properties.


ƒ'(t) = sin t + 2t; ƒ(0) = 5

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Step 1: Recognize that the problem involves finding the original function ƒ(t) given its derivative ƒ'(t) = sin(t) + 2t and an initial condition ƒ(0) = 5.
Step 2: To find ƒ(t), integrate the derivative ƒ'(t). Start by breaking ƒ'(t) into two terms: sin(t) and 2t. The integral of ƒ'(t) will be the sum of the integrals of these terms.
Step 3: Compute the integral of sin(t). Recall that the integral of sin(t) with respect to t is -cos(t).
Step 4: Compute the integral of 2t. Recall that the integral of t^n (where n = 1) is t^(n+1)/(n+1). Therefore, the integral of 2t is t^2.
Step 5: Combine the results from the integration: ƒ(t) = -cos(t) + t^2 + C, where C is the constant of integration. Use the initial condition ƒ(0) = 5 to solve for C by substituting t = 0 into ƒ(t).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It provides information about the function's slope at any given point. In this context, ƒ'(t) = sin t + 2t indicates how the function f changes as t varies, which is essential for finding the original function f.
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Derivatives

Integration

Integration is the process of finding the antiderivative of a function, essentially reversing differentiation. To find the function f from its derivative ƒ'(t), we need to integrate the expression sin t + 2t. This will yield a family of functions, which we can then refine using initial conditions.
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Integration by Parts for Definite Integrals

Initial Conditions

Initial conditions are specific values that help determine a unique solution to a differential equation or an integral. In this case, the condition ƒ(0) = 5 allows us to find the constant of integration after performing the integration, ensuring that the function f satisfies the given requirement at t = 0.
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Initial Value Problems
Related Practice
Textbook Question

104–107. Functions from derivatives Find the function f with the following properties.



h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3) 

Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]

Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = x³ - 6x² on [-1, 5]

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Textbook Question

45–46. Linear approximation


a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?


ƒ(x) = x²⸍³ ; a =27; ƒ(29)


Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

g(x) = x sin⁻¹ x on [-1, 1]

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Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


eˣ and 3ˣ