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Chapter 3: Problem 28

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 \(a\) $$f(x)=x^{2} ; a=3$$

Short Answer

Expert verified
Answer: The equation of the tangent line is \(y - 9 = 6(x - 3)\).

Step by step solution

01

Find the derivative of the function

Given that the function is \(f(x)=x^2\). To find the derivative of this function, we will use the power rule, which states \((x^n)' = nx^{n-1}\). Applying the power rule to \(f(x)=x^2\), we get: $$f'(x) = 2x$$
02

Evaluate the derivative at the given point

Now we need to find \(f'(a)\), where \(a=3\). To do this, substitute \(a\) into our derivative: $$f'(3) = 2(3) = 6$$
03

Calculate the value of the function at the given point

We also need the value of the function at \(a=3\), that is \(f(3)\). To evaluate the function at \(a=3\), substitute \(a\) into the function: $$f(3) = (3)^2 = 9$$ So, the point on the graph is \((3, 9)\).
04

Find the equation of the tangent line

Now, we will determine the equation of the tangent line at the point \((3, 9)\). To do this, recall that the point-slope form of the equation of a line is: $$y - y_{1} = m(x - x_{1})$$ Here, \((x_{1}, y_{1}) = (3,9)\) and the slope \(m = f'(3) = 6\). Substituting the values into the equation, we get: $$y - 9 = 6(x - 3)$$ This is the equation of the tangent line to the graph of \(f(x)=x^2\) at the point \((3,9)\).

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Most popular questions from this chapter

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