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Chapter 6: Problem 66

Find an antiderivative of \(f(x)\), call it \(F(x)\), and compare the graphs of \(F(x)\) and \(f(x)\) in the given window to check that the expression for \(F(x)\) is reasonable. [That is, determine whether the two graphs are consistent. When \(F(x)\) has a relative extreme point, \(f(x)\) should be zero; when \(F(x)\) is increasing, \(f(x)\) should be positive, and so on.] \(f(x)=2 x-e^{-.02 x},[-10,10]\) by \([-20,100]\)

Short Answer

Expert verified
Antiderivative: \(F(x) = x^2 - 50 e^{-0.02x} + C\). Check with derivatives and plot to ensure consistency.

Step by step solution

01

Find the antiderivative

The antiderivative of a function is found by integrating it. So, integrate the given function: \[F(x) = \int f(x) \, dx = \int (2x - e^{-0.02x}) \, dx \].
02

Integrate each term

Integrate the function term by term:1. For the term \(2x\): \[ \int 2x \, dx = x^2 + C_1 \], where \(C_1\) is a constant.2. For the term \(e^{-0.02x}\): \[ \int e^{-0.02x} \, dx = -50 e^{-0.02x} + C_2 \], where \(C_2\) is a constant.
03

Combine the results

Combine the results of the integrals obtained in Step 2 to get the form of the antiderivative: \[ F(x) = x^2 - 50 e^{-0.02x} + C \], where \(C\) is the combined constant of integration.
04

Determine the graph properties

To check whether \(F(x)\) is a reasonable antiderivative for \(f(x)\):1. Plot \(F(x) = x^2 - 50 e^{-0.02x} + C\) and \(f(x) = 2x - e^{-0.02x}\) in the given window \([-10, 10] \text{ by } [-20, 100]\).2. Ensure that when \(F(x)\) has a relative extremum (i.e., \(\frac{dF}{dx} = 0\)), \(f(x)\) should be zero.3. Check that, in regions where \(F(x)\) is increasing (i.e., \(\frac{dF}{dx} > 0\)), \(f(x)\) is positive, and where \(F(x)\) is decreasing (i.e., \(\frac{dF}{dx} < 0\)), \(f(x)\) is negative.
05

Verify with derivative

Differentiate \(F(x)\) and confirm it results in \(f(x)\): \[ \frac{d}{dx} [x^2 - 50 e^{-0.02x} + C] = 2x + 50 (0.02)e^{-0.02x} = 2x - e^{-0.02x} \]. This confirms the antiderivative is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
To find the antiderivative of a function, we need to integrate it. Integration can sometimes look tricky, but breaking it into simpler parts can help. For example, for the function given: \(f(x) = 2x - e^{-0.02x}\), we integrate each term separately.
For the first term, \(\frac{d}{dx}(x^2) = 2x\), so the integral of \(2x\) is \(\frac{2x^2}{2} = x^2\).
Next, the exponential term \(e^{-0.02x}\) can be integrated knowing \( \frac{d}{dx}(e^{kx}) = ke^{kx}\). We reverse this process to get: \( \frac{-1}{0.02}e^{-0.02x} = -50e^{-0.02x}\).
By combining them, we have \(F(x) = x^2 - 50e^{-0.02x} + C\), where \(C\) is a constant.
Function Analysis
A crucial step after finding an antiderivative is analyzing its behavior to check its correctness. Here's what we look for:
- If \(F(x)\) has a relative extremum (a minimum or maximum), \(\frac{dF}{dx}\) should be zero at that point.
- When \(F(x)\) is increasing, \(f(x)\) (which is \(\frac{dF}{dx}\)) should be positive. And when \(F(x)\) is decreasing, \(f(x)\) should be negative.
For our given \(F(x)\), we check these points by differentiating it back: \(F'(x) = 2x - e^{-0.02x}\), confirming it matches \(f(x)\). This analysis ensures our antiderivative behaves consistently with our original function.
Graphical Consistency
To visually compare \(F(x)\) and \(f(x)\), graph them in the specified window, \([-10,10] \text{ by } [-20,100]\).
First, plot \(f(x) = 2x - e^{-0.02x}\). This shows where our function rises or falls.
Next, plot \(F(x) = x^2 - 50e^{-0.02x} + C\). Ensure \(F(x)\)'s slope (how it rises or falls) matches \(f(x)\) across the interval.
- For instance, where \(F(x)\) rises, \(f(x) > 0\); where \(F(x)\) falls, \(f(x) < 0\).
- At \(F(x)\)'s peaks or valleys, \(f(x) = 0\).
This graphical comparison ensures our analytical work aligns with visual behavior, giving a concrete check of accuracy.

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