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Chapter 5: Problem 15

Consider the following functions \(f\) and real numbers a (see figure). a. Find and graph the area function \(A(x)=\int_{a}^{x} f(t) d t\) for \(f\) b. Verify that \(A^{\prime}(x)=f(x)\) $$f(t)=5, a=-5$$

Short Answer

Expert verified
The area function A(x) is given by A(x) = 5x + 25, and its derivative A'(x) is equal to 5. Additionally, we have verified that A'(x) = f(x), where f(x) = 5.

Step by step solution

01

Integrate f(t) to find A(x)

To find the area function \(A(x)\), we will integrate \(f(t)\) from \(a\) to \(x\). Since \(f(t)=5\), our integral will look like this: $$A(x) = \int_{-5}^{x} 5 dt$$ Now, we can integrate with respect to t: $$A(x) = \left[5t\right]_{-5}^x$$ Now evaluate the limits: $$A(x) = 5x - 5(-5) = 5x + 25$$
02

Graph the area function A(x)

Now, we will graph the area function \(A(x) = 5x + 25\), which is a linear equation. To graph this function, we can identify the slope and the y-intercept. The slope is \(5\) and the y-intercept is \((0, 25)\). So, the graph starts at the point \((0, 25)\) and has a positive slope of 5.
03

Calculate the derivative of the area function and verify A'(x)=f(x)

To verify that \(A'(x)=f(x)\), we need to calculate the derivative of \(A(x)\): $$A'(x) = \frac{d}{dx}(5x + 25)$$ Using basic differentiation rules, we have: $$A'(x) = 5$$ So, we have calculated the derivative \(A'(x)\) as \(5\). Additionally, our original function was defined as \(f(x) = 5\). Therefore, since \(A'(x)=5=f(x)\), we have verified the required condition.

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Most popular questions from this chapter

Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \sin \left(\pi t^{2}\right) d t \text { (a Fresnel integral) }$$

Multiple substitutions Use two or more substitutions to find the following integrals. $$\int x \sin ^{4}\left(x^{2}\right) \cos \left(x^{2}\right) d x$$ (Hint: Begin with \(u=x^{2}\), then use \(v=\sin u .)\)

Integral of \(\sin ^{2} x \cos ^{2} x\) Consider the integral \(I=\int \sin ^{2} x \cos ^{2} x d x\) a. Find \(I\) using the identity \(\sin 2 x=2 \sin x \cos x\) b. Find \(I\) using the identity \(\cos ^{2} x=1-\sin ^{2} x\) c. Confirm that the results in parts (a) and (b) are consistent and compare the work involved in each method.

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\sin x ; a=0, b=\pi / 2, c=\pi$$

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