Sample questions for test 2

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

Limits

• Graphically identify the following limits or say why they do not exist:

$$~ \lim_{x \rightarrow 0} \frac{\sqrt{x + 25} - 5}{x} ~$$$$~ \lim_{x \rightarrow 0} \frac{|x|}{x} ~$$$$~ \lim_{x \rightarrow 0} \frac{(3+x)^{-1} - (3-x)^{-1}}{x} ~$$$$~ \lim_{x \rightarrow 0+} \log(\frac{1}{|x|})^{-1/32} ~$$

• Using a table, identify the following limits. (You can also just look at $f(0.1), f(0.01), f(0.001), ...$)

$$~ \lim_{x \rightarrow 0} \frac{\sqrt{x + 25} - 5}{x} ~$$$$~ \lim_{x \rightarrow 0} \frac{|x|}{x} ~$$$$~ \lim_{x \rightarrow 0} \frac{(3+x)^{-1} - (3-x)^{-1}}{x} ~$$$$~ \lim_{x \rightarrow 0+} \log(\frac{1}{|x|})^{-1/32} ~$$

• Using SymPy find the following limits:

$$~ \lim_{x \rightarrow 0}\frac{\sqrt{1+x} - \sqrt{1-x}}{x}. ~$$$$~ \lim_{x \rightarrow 4} \frac{x^{3/2} - 8}{x-4} ~$$

(Use: f(x) = (x^(3//2) - 8) / (x - 4).)

$$~ \lim_{x \rightarrow 0} \frac{|x|}{x}. ~$$$$~ \lim_{x \rightarrow 0+} \log(\frac{1}{|x|})^{-1/32} ~$$

Derivatives

We assume you have run the following code:

using Roots
Base.ctranspose(f::Function) = D(f)

ctranspose (generic function with 40 methods)

• 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 functin $f(x) = \sin(x)$ 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?

• 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)?

• 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?