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.




  1. (3.4.31) Differentiate

    h(y) = $\displaystyle {\frac{{(1 + y)^{1/2} + (1 - y)^{1/2}}}{{y^{5/3}}}}$.

    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.

  2. Attempt to differentiate and check with .
    1. y = x13/2
    2. y = x13.3
    3. y = 13x

  3. 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) = ${\frac{{1}}{{f'(g(x))}}}$.
    1. f (x) = x2 -1,    x≥ 0
    2. f (x) = 1 + x2,    x≥ 0
    3. f (x) = $\sqrt{{x}}$ -1,    x≥ 0
    4. f (x) = $\sqrt[3]{{x}}$ -1,     - ∞ < x < + ∞

  4. For each curve, find all points on the graph where the tangent line is either horizontal or vertical. Check your work by using Plot.
    1. (3.4.36) y = x$\sqrt{{4 - x^2}}$
    2. (3.4.38) y = ${\frac{{1}}{{\sqrt{9 - x^2}}}}$

  5. 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.
    1. (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).
    2. (3.4.46) Find three distinct lines through the point P(3, 10) that are normal to the parabola y = x2.
    3. Generalize the above results. For what regions of the plane are there respectively 0, 1, 2, or 3 normals to y = x2?