Science:Math Exam Resources/Courses/MATH102/December 2011/Question 01 (b) iii
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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.
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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 ?
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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.
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: