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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.81
Chapter 4, Problem 4.7.81

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dv/dt = (1/2)sec t tan t, v(0) = 1

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1
Identify the given differential equation and initial condition: \(\frac{dv}{dt} = \frac{1}{2} \sec t \tan t\), with \(v(0) = 1\).
Recognize that this is a separable differential equation where the right side is a function of \(t\) only, so you can integrate both sides with respect to \(t\) to find \(v(t)\).
Set up the integral: \(v(t) = \int \frac{1}{2} \sec t \tan t \, dt + C\), where \(C\) is the constant of integration.
Recall the integral formula: \(\int \sec t \tan t \, dt = \sec t + C\). Use this to integrate the right side.
Apply the initial condition \(v(0) = 1\) to solve for the constant \(C\) by substituting \(t=0\) and \(v=1\) into the expression for \(v(t)\).

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Key Concepts

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

Differential Equations

A differential equation relates a function with its derivatives. Solving it involves finding the original function that satisfies the given relationship between the function and its rate of change.
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Classifying Differential Equations

Initial Value Problems (IVP)

An initial value problem specifies the value of the unknown function at a particular point, allowing for a unique solution to the differential equation by applying this initial condition.
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Initial Value Problems

Integration of Trigonometric Functions

Solving the differential equation requires integrating trigonometric expressions like sec t and tan t. Knowing standard integrals and techniques for these functions is essential to find the explicit solution.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Absolute Extrema on Finite Closed Intervals


In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.


f(x) = (2/3)x − 5, −2 ≤ x ≤ 3

Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x³ / (3x² + 1)

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Textbook Question

Theory and Examples


Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Textbook Question

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x² − 32√x

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

80. y' = 1 - cot²θ, for 0 < θ < π