Mathematics 133 – Intensive Calculus for Science 1 Discussion 5 – Derivative “Shortcuts,” I October 19, 2005

Background

Last time, we saw the derivative power rule: For all real n,

\frac{d}{dx}(x^{n}) = nx^{n−1}

and the Constant Multiple and Sum Rules: If f(x) and g(x) are differentiable at x and c is constant, then cf(x) and f(x) + g(x) are also differentiable at x and

\frac{d}{dx}(cf(x)) = cf′(x), \frac{d}{dx}(f(x) + g(x)) = f′(x) + g′(x).

Today, we want to practice using these to compute derivatives of polynomials and make use of the general facts about derivatives we know from Chapter 2.

Discussion Questions

A) Compute the derivatives of the following functions:

1) f(x) = 7x^{4} − 2x^{1/2} + 3x^{−2} + 1

2) g(x) = x^{4/3} − \frac{1}{8\sqrt{x}} + x

3) h(x) = x^{e} + πx^{\sqrt{2}} − \frac{1}{x^{π}} 4)

k(x) = \frac{5x^{2} − 2x + 3}{x^{1/3}}

(Hint: Split up into a sum of fractions and simplify. Then compute the derivative!)

B) Consider f(x) = x^{4} − 4x^{3}.

1) Find f′(x) and f′′(x) and factor each of them.

2) Find all x where f′(x) = 0, and determine the intervals where f′(x) > 0 and the intervals where f′(x) < 0.

3) Find all x where f′′(x) = 0, and determine the intervals where f′′(x) > 0 and the intervals where f′′(x) < 0.

4) Find the values of f(x) at the x’s from parts 2,3. Using those points and the other information from parts 2,3, sketch the graph y = x^{4} − 4x^{3}, indicating correct concavities.

C) Let P = (a, a^{2} − 2a + 4) be a general point on the parabola y = x^{2} − 2x + 4.

1) Find the equation of the tangent line to the parabola at P. (Give your answer in slope-intercept form. Both the slope and the intercept will depend on a, since the tangent lines are different depending on what a we pick.)

2) How many different a’s are there where the tangent line passes through (0, 0). Find them all.

3) Graph the parabola y = x^{2} − 2x + 4 together with the tangent lines passing through (0, 0).