Science:Math Exam Resources/Courses/MATH102/December 2015/Question 10 (a)

MATH102 December 2015

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Question 10 (a)
Consider the function $f(x)$ (solid curve) and its tangent line at $x=4$ (dashed). The graph of $f(x)$ is symmetric about $x=2$. Calculate the quantities below. (a) $\lim _{h\to 0}{\frac {f(4+h)-f(4)}{h}}$

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Hint
Recall the definition of the derivative.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the definition of a derivative, the given quantity equals to $f'(4)$; $f'(4)=\lim _{h\to 0}{\frac {f(4+h)-f(4)}{h}}$ To get $f'(4)$, it is enough to find the slope of the tangent line of the function $f$ at $x=4$. From the graph, we can see that the tangent line passes though the two points $(3,4)$ and $(4,0)$, so that its slope is ${\frac {0-4}{4-3}}=-4$. Therefore, we have $\color {blue}f'(4)=-4$