Chapter 3: Worksheet 5 Jack K. Cohen Colorado School of Mines
Generalized Power Rule; Inverse Functions
Suggested Problems
Section 3.4: 2, 4, 7, 10, 14, 20.
- (3.4.31) Differentiate
h(
y) =

.
Then use to check your work—you will probably need to do some additional hand simplifications to bring your answer into line with 's. In particular, use the command Together[h'[y]] to collect terms into a single fraction.
- Attempt to differentiate and check with .
-
y = x13/2
-
y = x13.3
- y = 13x
- For each of the following functions y = f (x) on the stated domain, (1) find the range and derivative. Then (2) find the inverse function y = g(x), (3) verify that g has the correct domain and range and (4) check that g'(x) =
.
-
f (x) = x2 -1, x≥ 0
-
f (x) = 1 + x2, x≥ 0
-
f (x) =
-1, x≥ 0
-
f (x) =
-1, - ∞ < x < + ∞
- For each curve, find all points on the graph where the tangent line is either horizontal or vertical. Check your work by using Plot.
- (3.4.36)
y = x
- (3.4.38) y =
- In the following problems, cubic equations arise. For the particular points mentioned, it is easy to guess a root and then use long division to reduce the problem to solving a quadratic. If your guessing and/or dividing powers are weak today, think about using tools like Plot, Table, Simplify and Factor to help out.
- (3.4.45) Find a line through the point P(18, 0) that is normal to the parabola y = x2 at some point Q(a, a2).
- (3.4.46) Find three distinct lines through the point P(3, 10) that are normal to the parabola y = x2.
- Generalize the above results. For what regions of the plane are there respectively 0, 1, 2, or 3 normals to y = x2?