Math 130-03, Spring 2001, Lab 7

Exercise 1

If a curve has a tangent line at a given point, then that tangent line is an approximation for the curve near that point. If you blow up the curve near the point by a large enough factor, it becomes indistinguishable from its tangent line. If $ f$ is a function that is differentiable at $ x=a$, and if you blow up the curve near the point $ (a,f(a))$, you will see that the curve starts to look linear at that point. That is, on a small enough scale, the curve looks like its tangent line. This means that it is possible to estimate $ f'(a)$ by blowing up the curve near the point $ (a,f(a))$ until the curve looks like a line. You can then simply estimate the slope of that line. Of course, this depends on the fact that $ f'(a)$ exists. If it doesn't exist, then no matter how much you blow up the graph near $ (a,f(a))$, it will not become linear.

The following applet has four different examples available. You can view an example by selecting it from the pop-up menu at the top of the applet and then clicking the ``Load Example'' button. Your assignment is to investigate $ f'(1)$ for each of the four examples by blowing up the curve repeatedly near the point $ (1,f(1))$. You can mark this point on the graph by entering 1 in the box at the bottom of the applet. This will help you make sure that you keep the point in view as you magnify the graph. You can zoom in on a point by clicking it. When visually estimating a slope, it's nice to have the scales on the x-axis and y-axis be the same. Click the ``Equalize Axes'' button to make sure that the scales are the same. The grid lines can help you to get a good estimate of the slope.

Turn in your estimate for $ f'(1)$ for each example where this derivative exists, and explain briefly how you got your answers. In one example, $ f'(1)$ does not exist. In that case, explain how you can tell that this derivative does not exist.

Exercise 2

We have seen rules for finding the derivative of a sum, difference, product, or quotient of two functions. But the most important operation on functions is composition. The chain rule is used to find the derivative of a composition of two functions. It says that if $ h(x)=g(f(x))$, then $ h'(x)=g'(f(x))f'(x)$. We will soon encounter this rule in class. A special case of the chain rule is the generalized power rule, which says: if $ f$ is a function and if $ c$ is a constant, then

$\displaystyle {\frac{d}{dx}}\left(f(x)^c\right)=cf(x)^{c-1}{\frac{d}{dx}}f(x)$

This exercise discusses a proof of this rule, in the case when $ c$ is a integer. You will need to exercise your algebra as well as your reasoning skills for this one:

(a) Note that $ f(x)^2$ can be written as a product, $ f(x)\cdot f(x)$. Use this fact and the product rule to compute that $ {\frac{d}{dx}}\left(f(x)^2\right)=2f(x)^{2-1}{\frac{d}{dx}}f(x)$.

(b) Note that $ f(x)^3$ can be written as a product, $ f(x)\cdot f(x)^2$. Use this fact, the product rule, and the result from part (a) to compute that $ {\frac{d}{dx}}\left(f(x)^3\right)=3f(x)^{3-1}{\frac{d}{dx}}f(x)$.

(c) Suppose that $ n$ is a positive integer. Suppose that it has already been proven that $ {\frac{d}{dx}}\left(f(x)^n\right)=nf(x)^{n-1}{\frac{d}{dx}}f(x)$ Note that $ f(x)^{n+1}$ can be written as a product, $ f(x)\cdot f(x)^n$. Use these facts and the product rule to compute that $ {\frac{d}{dx}}\left(f(x)^{n+1}\right)=(n+1)f(x)^{(n+1)-1}{\frac{d}{dx}}f(x)$.

(d) Try to explain why parts (a), (b), and (c) taken together prove that $ {\frac{d}{dx}}\left(f(x)^k\right)=kf(x)^{k-1}{\frac{d}{dx}}f(x)$ for any positive integer $ k$. (Hint: Why do (b) and (c) together prove that $ {\frac{d}{dx}}\left(f(x)^4\right)=4f(x)^{4-1}{\frac{d}{dx}}f(x)$? Then why does this fact, together with (c) prove that $ {\frac{d}{dx}}\left(f(x)^5\right)=5f(x)^{5-1}{\frac{d}{dx}}f(x)$?)

(e) Suppose that $ n$ is a positive integer. Use the previous result and the quotient rule to show that $ {\frac{d}{dx}}\left(f(x)^{-n}\right)={-n}f(x)^{-n-1}{\frac{d}{dx}}f(x)$.

(f) Use the generalized power rule to compute $ \frac{d}{dx}\big(2x^2-3\big)^{17}$.

Exercise 3

The rules that we have seen for differentiating formulas include the Constant Multiple Rule, the Sum/Difference Rule, the Product Rule, the Quotient Rule, and (from the previous exercise) the Generalized Power Rule. The key to successfully differentiating formulas is to apply the rules one at a time, in a step-by-step fashion. Here are some derivatives that you can not do at this time. However, in each case, you can do one step in the problem by applying one of the above rules. In each case, show one step in the differentiation of the formula, and state which of the above rules you are applying. ($ \ln(x)$ is the so-called ``natural logarithm function,'' and $ \arcsin(x)$ is the so-called ``inverse sine function.'' You do not need to know anything about these functions to do this problem.)

(a) $ \displaystyle \frac{d}{dx}\,x^22^x$

(b) $ \displaystyle \frac{d}{dt}\,\left(\frac{\sin(t)\cos(t)}{2^t+2^{-t}}\right)$

(c) $ \displaystyle \frac{d}{dx}\,\root 3 \of{\ln\big(3+\sin(x)\big)\,}$

(d) $ \displaystyle \frac{d}{dz}\,\left(\frac{z}{3z-1}\cdot\frac{z^2+1}{\arcsin(z)}\right)$

(e) $ \displaystyle \frac{d}{dx}\big(\sin(x)+2^x\cos(x)\big)$

David Eck, March 2001