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Chapter 2: Problem 7

Find the slope of the tangent line to the graph of the function at the given point. \(g(x)=x^{2}-4, \quad(1,-3)\)

Short Answer

Expert verified
The slope of the tangent line to the graph \(g(x) = x^2 - 4\) at the point (1, -3) is equal to 2.

Step by step solution

01

Finding the Derivative of \(g(x)\)

We'll need to find the derivative of the given function. In this case, \(g(x) = x^2 - 4\). The derivative of \(x^2\) is \(2x\) and the derivative of \(4\) is \(0\). So, the derivative of \(g(x)\) is \(2x\).
02

Evaluating the Derivative at a Specific Point

Now that we have the derivative \(g'(x) = 2x\), we need to find the slope of the tangent line at the specific point. This is the \(x\)-coordinate of the point, \(x = 1\). Put \(1\) into the derivative function, which yields \(g'(1) = 2*1 = 2\). So the slope of the tangent line at the point (1, -3) is 2.
03

Confirming the Point lies on the given Function

Lastly, we check if the point (1, -3) actually lies on the graph of \(g(x) = x^2 - 4\). When we replace \(x\) with \(1\), we get \(g(1) = 1^2 - 4 = -3\). Luckily, this value matches with the \(y\)-coordinate from the given point, confirming that the point is on the function. Therefore, the slope that we found in step 2, is indeed correct.

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Most popular questions from this chapter

Find equations of both tangent lines to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) that passes through the point (4,0).

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