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Chapter 3: Problem 27

Derivatives and tangent lines a. For the following functions and values of \(a,\) find \(f^{\prime}(a)\) b. Determine an equation of the line tangent to the graph of \(f\) at the point \((a, f(a))\) for the given value of \(a\) $$f(x)=8 x ; a=-3$$

Short Answer

Expert verified
In summary, the tangent line to the function \(f(x) = 8x\) at the point \(a = -3\) has a slope of \(8\) and passes through the point \((-3, -24)\). The equation of the tangent line is \(y = 8x\).

Step by step solution

01

Find the derivative of the function:

To find the derivative of the function \(f(x) = 8x\), we need to differentiate \(f(x)\) with respect to \(x\). Using the power rule, the derivative of \(x^n\) is \(nx^{n-1}\), so: $$f^{\prime}(x) = 8 (1) x^{1-1} = 8$$
02

Evaluate the derivative at a:

Now we need to find the value of \(f^{\prime}(a)\) where \(a = -3\). Since the derivative is a constant value \(8\), the value of \(f^{\prime}(-3)\) is: $$f^{\prime}(-3) = 8$$
03

Find the point on the graph:

To find the point on the graph of the function \(f(x)\), we need to find the value of \(f(-3)\): $$f(-3) = 8(-3) = -24$$ So the point on the graph is \((-3, -24)\).
04

Determine the equation of the tangent line:

We've found that the slope of the tangent line is \(8\) and the point on the graph is \((-3, -24)\). Using the point-slope form of a line, we have \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is the point on the graph. Substitute the values to get the equation of the tangent line: $$y - (-24) = 8(x - (-3))$$ Simplifying the equation will give us the equation of the tangent line: $$y + 24 = 8(x + 3)$$ $$y = 8x + 24 - 24$$ $$y = 8x$$ The equation of the tangent line is \(y = 8x\).

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