calculus homework

MTH 251 Lab 7 FALL 2016

NAME :

1. Inverse trigonometric derivatives review. Find the derivative of the following functions:

(a) d dx

(ln(tan−1(x2 + 3))). For what values of x is this derivative valid?

(b) d dx

(π sec−1(x2 − 1)). For what values of x is this derivative valid?

(c) d dx

( 11 sin−1(−3x)

) . For what values of x is this derivative valid?

2. Consider the linear function y = 1 − 2x. Find the inverse function, then compute the derivatives of both functions. What do you notice?

3. The relationship in the previous problem works for any differentiable function with an inverse. If f is differentiable at a point x0, then the derivative of f

−1 at an appropriate point should be 1/f ′(x0), if f

′(x0) 6= 0.

(a) Sketch the graph of y = √ x− 1, then find and carefully sketch its inverse on the

same graph.

(b) Find the tangent line to the graph of y = √ x− 1 at the point where x = 5.

Sketch the tangent line on your curve.

(c) What is the corresponding tangent line on the graph of the inverse function? What is its slope? You should be able to get this just by thinking about the geometry.

(d) Compute the derivative of the inverse function directly. What input value must you plug-in to get 1/y′(5)? (This is the slope you should have found in the previous part.)

4. The previous problem illustrates the following result: If f is differentiable and has an inverse on some interval I, and x0 is a point in I such that f

′(x0) 6= 0, then f−1 is differentiable at y0 = f(x0), and

(f−1)′(y0) = 1

f ′(x0) .

Use this to determine (f−1)′(3) for the function f(x) = x3 + x + 1, without finding the inverse first.