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Chapter 4: Problem 37

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=2 x+1$$

Short Answer

Expert verified
Question: If the given function is \(f(x) = 2x + 1\), determine the relationship between a small change in \(x\) (denoted as \(dx\)) and the corresponding change in \(y\) (denoted as \(dy\)). Answer: The relationship between \(dy\) and \(dx\) for the function \(f(x) = 2x + 1\) is \(dy = 2 dx\).

Step by step solution

01

Find the derivative of the function f(x)

In this case, the function is \(f(x) = 2x + 1\). To find the derivative, we differentiate \(f(x)\) with respect to \(x\), getting: $$f'(x) = \frac{d(2x + 1)}{dx}$$ Since the derivative of \(2x\) with respect to \(x\) is \(2\) and the derivative of the constant \(1\) with respect to \(x\) is zero, we have: $$f'(x) = 2$$
02

Relate the change in y (dy) to the change in x (dx)

Now that we have obtained the derivative \(f'(x) = 2\), we can relate the changes in \(y\) and \(x\) as follows: $$dy = f'(x) dx$$ Substitute the expression for \(f'(x)\) in the equation: $$dy = 2 dx$$ This relationship implies that the change in \(y\) is twice the change in \(x\).

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Most popular questions from this chapter

Consider the functions \(f(x)=\frac{1}{x^{2 n}+1},\) where \(n\) is a positive integer. a. Show that these functions are even. b. Show that the graphs of these functions intersect at the points \(\left(\pm 1, \frac{1}{2}\right),\) for all positive values of \(n\) c. Show that the inflection points of these functions occur at \(x=\pm \sqrt[2 n]{\frac{2 n-1}{2 n+1}},\) for all positive values of \(n\) d. Use a graphing utility to verify your conclusions. e. Describe how the inflection points and the shape of the graphs change as \(n\) increases.

The graph of \(f^{\prime}\) on the interval [-3,2] is shown in the figure. a. On what interval(s) is \(f\) increasing? Decreasing? b. Find the critical points of \(f .\) Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does \(f\) have an inflection point? d. On what interval(s) is \(f\) concave up? Concave down? e. Sketch the graph of \(f^{\prime \prime}\) f. Sketch one possible graph of \(f\)

Evaluate one of the limits l'H么pital used in his own textbook in about 1700: \(\lim _{x \rightarrow a} \frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}},\) where \(a\) is a real number.

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\). a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A payload is dropped at an elevation of \(400 \mathrm{m}\) from a hot-air balloon that is descending at a rate of \(10 \mathrm{m} / \mathrm{s}\).

Suppose \(f(x)=1 /(1+x)\) is to be approximated near \(x=0\). Find the linear approximation to \(f\) at 0 . Then complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is \(100 \cdot |\) approximation \(-\) exact \(|/|\) exact \(| .\) Comment on the behavior of the errors as \(x\) approaches 0 .
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