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Chapter 5: Problem 20

Consider the following functions \(f\) and real numbers a (see figure). a. Find and graph the area function \(A(x)=\int_{a}^{x} f(t) d t\) b. Verify that \(A^{\prime}(x)=f(x)\) $$f(t)=2 t+5, a=0$$

Short Answer

Expert verified
Answer: The area function, A(x), is given by A(x) = x^2 + 5x. Upon verifying the fundamental theorem of calculus, we found that A'(x) = 2x + 5, which is equal to the given function f(x).

Step by step solution

01

Integrate the given function

To find the area function, \(A(x)\), we need to integrate the given function, \(f(t) = 2t + 5\), with respect to \(t\) from \(0\) to \(x\): \(A(x) = \int_0^x f(t) dt = \int_0^x (2t + 5) dt\)
02

Calculate the indefinite integral

Now, we will find the indefinite integral of \((2t + 5)\) with respect to \(t\): \(\int (2t + 5) dt = t^2 + 5t + C\)
03

Compute the area function

Since we have already found the indefinite integral, we can compute the definite integral from \(0\) to \(x\): \(A(x) = \int_0^x (2t + 5) dt = (x^2 + 5x + C) - (0^2 + 5\cdot0 + C) = x^2 + 5x\) Now, we have the area function \(A(x) = x^2 + 5x\). To graph this function, we must understand that it is a quadratic function with a positive leading coefficient, so it will have a parabolic shape, opening upward. #b. Verify that \(A'(x) = f(x)\).#
04

Differentiate the area function

To verify the fundamental theorem of calculus, we need to find the derivative of the area function, \(A'(x)\), with respect to \(x\): \(\frac{d}{dx} (A(x)) = \frac{d}{dx}(x^2 + 5x) = 2x + 5\)
05

Verify the result

Now, we will verify if our result is indeed equal to \(f(x)\): \(A'(x) = 2x + 5 = f(x)\) We can see that the derivative of the area function \(A'(x)\) is indeed equal to the given function \(f(x)\). We have successfully verified the fundamental theorem of calculus.

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Most popular questions from this chapter

General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. $$\int 2\left(f^{2}(x)+2 f(x)\right) f(x) f^{\prime}(x) d x$$

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. $$g(x)=\int_{0}^{x} \sin \left(\pi t^{2}\right) d t \text { (a Fresnel integral) }$$

Find the area of the following regions. The region bounded by the graph of \(f(x)=x \sin \left(x^{2}\right)\) and the \(x\) -axis between \(x=0\) and \(x=\sqrt{\pi}\).

Consider the right triangle with vertices \((0,0),(0, b),\) and \((a, 0)\) where \(a>0\) and \(b>0 .\) Show that the average vertical distance from points on the \(x\) -axis to the hypotenuse is \(b / 2,\) for all \(a>0\).

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. $$g(x)=\int_{0}^{x}\left(t^{2}+1\right) d t$$
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