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Chapter 7: Problem 23

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 \sqrt{x} ;(4,4) $$

Short Answer

Expert verified
The slope of the tangent line to the graph of \(f(x) = 2 \sqrt{x}\) at the point (4, 4) is \(1/2\).

Step by step solution

01

Substituting into the equation

Substitute \(f(a + h)\) and \(f(a)\) into the equation. We have \(a = 4\) and \(f(x) = 2 \sqrt{x}\), calculating these gives: \(f(a + h) = 2 \sqrt{4 + h}\) and \(f(a) = 2 \sqrt{4}\).
02

Solve the equation

Now that we have \(f(a + h)\) and \(f(a)\), we can substitute those into the definition of the derivative and solve for \(f'(a)\). Plug these values into the equation: \[f'(4) = \lim_{h\to 0} \[ \frac{2 \sqrt{4 + h} - 2 \sqrt{4}}{h} \]\]
03

Simplify the equation

Simplify the equation further for easy computation. This simplification can be achieved by rationalizing the numerator. The result is: \[f'(4) = \lim_{h\to 0} \[ \frac{2}{\sqrt{4 + h} + 2} \]\]
04

Compute the limit

Since we have simplified the equation, we can easily compute the limit as \(h\) approaches 0. Simply replace \(h\) with 0 in the equation. The result is: \(f'(4) = 1/2\)

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Most popular questions from this chapter

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=x \sqrt{x^{2}+5} $$

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{8^{2}} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{1}{\left(x^{2}-3 x\right)^{2}} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=3-g(x) $$
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