Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.7.19

Capitolo 4, Problema 4.7.19

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(3t² + t/2) dt

Guida verificata passo dopo passo

Identify the integral to solve: \(\int \left(3t^{2} + \frac{t}{2}\right) \, dt\).

Split the integral into the sum of two separate integrals: \(\int 3t^{2} \, dt + \int \frac{t}{2} \, dt\).

Apply the power rule for integration to each term separately. Recall that \(\int t^{n} \, dt = \frac{t^{n+1}}{n+1} + C\) for \(n \neq -1\).

Integrate the first term: \(\int 3t^{2} \, dt = 3 \int t^{2} \, dt = 3 \cdot \frac{t^{3}}{3} = t^{3}\).

Integrate the second term: \(\int \frac{t}{2} \, dt = \frac{1}{2} \int t \, dt = \frac{1}{2} \cdot \frac{t^{2}}{2} = \frac{t^{2}}{4}\), then combine the results and add the constant of integration \(C\).

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Indefinite Integral

An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C. It does not have specified limits, unlike definite integrals.

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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is fundamental for integrating polynomial terms like t² or t/2 by increasing the exponent by one and dividing by the new exponent.

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Power Rule for Indefinite Integrals

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral solution and helps identify any errors in the integration process.

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Finding Differentials

Pratica correlata

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In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

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The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

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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) = x¹ᐟ³(x + 8)

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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.

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Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 1/x² + x, x > 0; y(2) = 1

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