Sample questions for test 3

There may be other types of questions, but being able to do these should mean you will do really well on this test.

Derivatives

We assume you have run the following code:

using MTH229

• Find the slope of the secant line of the function $f(x) = \sin(x)$ between $-\pi/3$ and $\pi/3$.

• Find the slope of the tangent line of the function $f(x) = \sin(x)$ at both $-\pi/3$ and $\pi/3$

• Let $h=10^{-6}$. Find the approximate forward derivative of $f(x) = \sin(x)$ at $\pi/3$.

• What is the difference in value between your last number and that given by sin'(pi/3)?

• An arrow is launched. It's height as a function of time is given by:

$$~ y(t) = -16t^2 + 25t + 5, \quad t \geq 0 ~$$

Find the rate of change of height at time $t=1$. Find the rate of change of height at the instant the arrow strikes the grount.

• Let $V(t) = 10(1 - t/100)^2$. Find the averate rate of change between $t=0$ and $t=100$. Call this value $a$. Using fzero find when $V'(t) = a$ between $0$ and $100$.

First and second derivatives

• Plot the function f(x) = x * airy(x) over the interval $[-3, 3]$. From your graph, answer: on what intervals is $f(x)$ positive? increasing? concave up? (Use plotif here.)

• Find the critical point(s) of $f(x) = 2x - x^2/6 - x^3/9$. What are the values of $f''(x)$ at these point(s)? (Use fzero or fzeros here.)

• Find the critical point(s) of $f(x) = 1/x + x, \quad x > 0$. What are the values of $f''(x)$ at these point(s)?

• Find any values where the second derivative of $f(x) = 2x - x^2/6 - x^3/9$ is $0$.

• Let $f(x) = x^3 - 3x + 5$. Find the critical points. Use the first derivative test to classify them as relative maxima, relative minima, or neither.

• Let $f(x) = x^2 e^{-x/3}$. Find the critical points. Use the first derivative test to classify them as relative maxima, relative minima, or neither.

• For the function f(x) = (x^2)^(1/3)*(2-x). Make a plot over $[-4,4]$ and identify graphically any critical points. Are all the critical points found by fzeros(f, -4, 4)? Can the second derivative test classify all the critical points? Can the first derivative test classify all the critical points?