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Chapter 2: Problem 8

Find the slope of the tangent line to the graph of the function at the given point. \(g(x)=5-x^{2}, \quad(2,1)\)

Short Answer

Expert verified
The slope of the tangent line to the function at the given point is -4.

Step by step solution

01

Obtain the Derivative of the Function

The derivative of the function \( g(x) = 5-x^{2} \) can be obtained by applying the power rule for differentiation. The power rule states that the derivative of \( x^n \) where \( n \) is a constant is \( n \cdot x^{(n-1)} \). So, the derivative of \( g(x) \) is \( g'(x)= -2x \).
02

Evaluate the Derivative at the Given Point

Substitute the x-coordinate of the given point into the derivative to obtain the slope of the tangent line at that point. So the slope \( m \) is \( g'(2) = -2(2) = -4 \).
03

State the Slope

Thus the slope of the tangent line to the function \( g(x) = 5-x^{2} \) at the point (2,1) is -4.

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Most popular questions from this chapter

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(f(x)=\cos \left(x^{2}\right), \quad(0,1)\)

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{f(x)=\frac{1}{3} x \sqrt{x^{2}+5}} \quad \frac{\text { Point }}{(2,2)}\)

Let \(f\) be a differentiable function of period \(p\). (a) Is the function \(f^{\prime}\) periodic? Verify your answer. (b) Consider the function \(g(x)=f(2 x)\). Is the function \(g^{\prime}(x)\) periodic? Verify your answer.

Find the derivative of the function. \(y=\log _{3} x\)

Find the derivative of the function. \(f(t)=t^{3 / 2} \log _{2} \sqrt{t+1}\)
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