Science:Math Exam Resources/Courses/MATH102/December 2017/Question 05

MATH102 December 2017

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Question 05
Below is the a graph of an exponential function ${\textstyle f(x)=a^{x}}$ with base ${\textstyle a}$. Which of the following values is closest to ${\textstyle \ln(a)}$? A. ${\textstyle -1}$ Template:0 B. ${\textstyle 0}$ Template:0 C. ${\textstyle 1}$ Template:0 D. ${\textstyle 1.5}$Template:0 E. ${\textstyle 3}$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Consider the derivative of ${\textstyle f(x)}$.

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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 Hint, let us consider the derivative of the function at ${\textstyle x=0}$. In fact, $$f'(x)=(a^{x})'=a^{x}\ln(a).$$ Therefore, $$f'(0)=a^{0}\ln(a)=\ln(a).$$ So we need to approximate ${\textstyle f'(0)}$. Geometrically, ${\textstyle f'(0)}$ is the slope of the tangent line at ${\textstyle x=0}$. This tangent line is approximated by a secant line joining, say, ${\textstyle (0,1)}$ and ${\textstyle (0.1,1.3)}$. So $$\ln(a)=f'(0)\approx {\dfrac {1.3-1}{0.1-0}}={\dfrac {0.3}{0.1}}=3.$$ Answer: ${\textstyle {\color {blue}E}}$.