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Chapter 2: Problem 4

Describe a process for finding the slope of the line tangent to the graph of \(f\) at \((a, f(a))\)

Short Answer

Expert verified
Answer: To find the slope of the tangent line at the point (a, f(a)), calculate the derivative of the function, f'(x), and evaluate it at the point a by substituting a into the derivative, giving f'(a). The result, f'(a), is the slope of the tangent line at the point (a, f(a)).

Step by step solution

01

Find the derivative of f(x)

To find the slope of the tangent line, we first need to find the derivative of the function, \(f'(x)\). Unfortunately, the given problem does not provide the function, we will refer to the derivative as \(f'(x)\).
02

Evaluate the derivative at point a

Once we have found the derivative of the function, \(f'(x)\), we need to evaluate it at the given point, \(a\). This can be done by plugging in \(a\) into the derivative function as \(f'(a)\).
03

Interpret the result

The result from step 2, \(f'(a)\), represents the slope of the tangent line to the graph of \(f\) at the point \((a, f(a))\). This slope indicates the rate of change of the function at that specific point.

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Most popular questions from this chapter

Prove Theorem 11: If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a),\) then the composition \(f \circ g\) is continuous at \(a .\) (Hint: Write the definition of continuity for \(f\) and \(g\) separately; then, combine them to form the definition of continuity for \(\left.f^{\circ} g .\right)\)

Determine the value of the constant \(a\) for which the function $$f(x)=\left\\{\begin{array}{ll} \frac{x^{2}+3 x+2}{x+1} & \text { if } x \neq-1 \\\a & \text { if } x=-1\end{array}\right.$$ is continuous at \(-1.\)

Use the continuity of the absolute value function (Exercise 62 ) to determine the interval(s) on which the following functions are continuous. $$h(x)=\left|\frac{1}{\sqrt{x}-4}\right|$$

Evaluate the following limits. $$\lim _{x \rightarrow 3 \pi / 2} \frac{\sin ^{2} x+6 \sin x+5}{\sin ^{2} x-1}$$

a. Sketch the graph of a function that is not continuous at \(1,\) but is defined at 1. b. Sketch the graph of a function that is not continuous at \(1,\) but has a limit at 1.
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