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Chapter 2: Problem 80

Use a graphing utility, with a square window setting, to zoom in on the graph of \(f(x)=4 \sqrt{x}+1\) to approximate \(f^{\prime}(4)\). Use the derivative to find \(f^{\prime}(4)\).

Short Answer

Expert verified
The approximate value of \(f^{\prime}(4)\) obtained by zooming in on the graph should be close to the exact derivative at \(x=4\), which is 4.

Step by step solution

01

Graph the function

Using your graphing utility, input the function \(f(x)=4\sqrt{x}+1\). Make sure to set it in a square window mode to accurately visualize the function.
02

Zoom in on the graph at \(x=4\)

Use the zoom feature of your graphing tool to focus on the region around \(x=4\). By doing this, you are getting a close view of the behaviour of the function at that specific point.
03

Approximate the derivative

The derivative at a point on the graph is equal to the slope of the tangent line at that point. So, observe the slope of the function at \(x=4\) and approximate the derivative as best as you can.
04

Find the derivative using calculus

The derivative of \(f(x)=4\sqrt{x}+1\) can be calculated using the power rule for differentiation that states: if \(f(x)=x^n\), then \(f^{\prime}(x)=nx^{n-1}\). The derivative \(f^{\prime}(x)\) of \(4\sqrt{x}+1\) is then \(2\sqrt{x}\). Substitute \(x=4\) into \(f^{\prime}(x)\) to get \(f^{\prime}(4) = 2\sqrt{4} = 4\).

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Most popular questions from this chapter

(a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. $$ g(t)=\frac{3 t^{2}}{\sqrt{t^{2}+2 t-1}}, \quad\left(\frac{1}{2}, \frac{3}{2}\right) $$

(a) find the specified linear and quadratic approximations of \(f,(b)\) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). $$ f(x)=\sec 2 x $$

Find \(d^{2} y / d x^{2}\) in terms of \(x\) and \(y\). $$ y^{2}=x^{3} $$

Think About It Describe the relationship between the rate of change of \(y\) and the rate of change of \(x\) in each expression. Assume all variables and derivatives are positive. (a) \(\frac{d y}{d t}=3 \frac{d x}{d t}\) (b) \(\frac{d y}{d t}=x(L-x) \frac{d x}{d t}, \quad 0 \leq x \leq L\)

In your own words, state the guidelines for implicit differentiation.
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