MATH102 December 2011
• Q1 (a) i • Q1 (a) ii • Q1 (b) i • Q1 (b) ii • Q1 (b) iii • Q1 (c) • Q2 (a) i • Q2 (a) ii • Q2 (a) iii • Q2 (b) i • Q2 (b) ii • Q2 (b) iii • Q2 (c) • Q3 • Q4 (i) • Q4 (ii) • Q4 (iii) • Q4 (iv) • Q4 (v) • Q4 (vi) • Q5 • Q6 • Q7 (i) • Q7 (ii) • Q7 (iii) • Q8 (i) • Q8 (ii) • Q8 (iii) • Q9 •
Question 01 (b) iii
Short-Answer Questions. A correct answer in the box gives full marks. For partial marks work needs to be shown.
Let be the inverse function of . Assume and Find the tangent line to at 1.
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!
The standard form of a line is given by
If the above line lies tangent to the function at , what is ?
If the above line must touch the function at a point, what are the values of ?
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
To determine the equation of the tangent line to at the point we need two things: (1) the slope of at and (2) the value of a point that the tangent line crosses through.
(1) Recall this important property of inverse functions:
From this rule, we can use implicit differentiation to get the derivative for in terms of .
Thus, the slope of the inverse function at the point is given by where
We know that the tangent line must touch the inverse function at the point . Using the rules of inverse functions, since , we know that , and thus
(2) We get by the above point as well: .
Using these two things in the standard equation of a line, we get an equation for the tangent line
Rearranging the equation to slope-intercept form (as the question asks), we get our final answer:
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Implicit differentiation, MER Tag Tangent line, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag