Science:Math Exam Resources/Courses/MATH110/April 2018/Question 01 (c)

MATH110 April 2018

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Question 01 (c)
Find the equation of the tangent line to ${\textstyle y=x^{3}}$ at ${\textstyle x=2}$ .

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
To find the equation of a line, you need to find two things: the slope of the line and a point contained on the line.

Hint 2
The slope of the tangent line to ${\textstyle y=f(x)}$ at ${\textstyle x=a}$ is equal to ${\textstyle f'(a)}$ .

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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. The first step will be to find the slope of the tangent line to ${\textstyle y=x^{3}}$ at ${\textstyle x=2}$ , which is given by the derivative of ${\textstyle y=x^{3}}$ at ${\textstyle x=2}$ . Using the power rule, we see that the derivative of ${\textstyle x^{3}}$ is ${\textstyle 3x^{2}}$ , so the derivative of ${\textstyle y=x^{3}}$ at ${\textstyle x=2}$ is ${\textstyle 3\cdot 2^{2}=12}$ . We want to use the slope-point formula to find the equation of the tangent line. Having found the slope, it remains to find a point contained on the line. The tangent line must pass through ${\textstyle y=x^{3}}$ at ${\textstyle x=2}$ , so it must contain the point ${\textstyle (2,2^{3})=(2,8)}$ . Now the slope-point formula tells us that the equation of the tangent line is given by ${\textstyle y-8=12(x-2)}$ . After rearranging, we get the equation ${\textstyle y=12x-16}$ . Answer: The equation of the tangent line at ${\textstyle x=2}$ is ${\textstyle {\color {blue}y=12x-16}}$ .