Exercises - Calculating Derivative: Inverse, Implicit and Parametric Functions

We start with problems on derivative of inverse functions, then do some on implicit functions and differentiation including tangent lines. Next we ask about derivatives and tangent lines for parametric curves and as a bonus we look at some angles of intersection.

If you want to refer to sections of Methods Survey on implicit and parametric functions while working the exercises, you can click here and it will appear in a separate full-size window. Similarly, here we offer Theory.

 

For each of the following functions f and a given point b, guess some a such that f (a) = b and then prove that f is 1-1 on some neighborhood of a and therefore it has an inverse function f−1 there. Then find ( f−1)′(b), the derivative of this inverse at b.

For each of the following curves, show that it can be given as a graph of some function y = y(x) on a neighborhood of the given point (a,b). Then use implicit differentiation to find y′(a) and y′′(a) and find the tangent line to this curve at (a,b).

For each of the following parametric curves, find the tangent line and the normal line at the indicated point.

The following parametric curves can be locally expressed as graphs of functions y = y(x). Find the derivatives y′(x) and y′′(x).

Bonus: Find the angles at which the following pairs of curves intersect.

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