Some sample questions for test 1.

calculator

Evaluate each expression. For each, write the answer and the julia commands you used to produce the answer:

• $10^9 / 8 \cdot 7 - 6 + 5$

• $\sin(\pi/3) \cdot \sin^2(\pi/4) \cdot \sin^{-1}(\pi/5)$

• $\cos(75^\circ) - \cos(75)$

• $e^{1/2 \cdot(3 - 1)^2}$

*

$$~ \frac{\sin(x) + \cos(x)}{\sin(x) - \cos(x)}, \text{ when } x = 1. ~$$

*

$$~ \sin^2 \left(\frac{x - 2\pi}{x + 2\pi}\right) \text{ when } x = \pi/4. ~$$

Order of operations

Which of these three expressions will be different from the other two?

• 2/3/4, (2/3)/4, 2/(3/4)

• 4^3^2, (4^3)/2, 4^(3^2)

• 2 - 3 - 4, (2-3)-4, 2 - (3-4)

Integers can cause some issues.

Are these expressions identical

• 2^(-1), 2.0^(-1)

• 2^64, 0

Floating point can cause small issues.

Are these expressions identical?

• .1 + .2 - .3, 0

• sin(pi), 0

• cos(pi), -1.0

Defining functions

For each function, write a function in Julia and compute $f(1/2)$. For each, your answer should have the command to define the function and the value at $1/2$.

• $f(x) = x^4 - 4x^2 + 2$

• $f(x) = \sqrt{x / (x + 1)}$

• $f(x) = \ln(x + 1) - x$

• $f(x) = \cos^{-1}(x)$

*

$$~ f(x) = \begin{cases} x^2 & x < 1\\ x & \text{otherwise} \end{cases} ~$$

• A function describing the line that goes through the points $(-1,-\pi)$ and $(2,\pi)$.

Plotting

• Graph the function f(x) = x * (x-1) / (x -2) / (x-3). Are there vertical asymptotes? Is there a slant asymptote? Is there a horizontal asymptote?

• Graph the function f(x) = cos(x) * cosh(x) between $[1, 5]$. Identify from the graph how many values of $x$ satisfy $f(x) = 1$ in $[1,5]$? What is the largest of them?

• Graph the functions $f(x) = \sin(x)$ and $g(x) = x - x^3/6$ over the interval $[-\pi/2, \pi/2]$. Can you tell how many times the two functions cross? Does it help to plot $\sin(x) - (x - x^3/6)$? Would it be helpful to know that $\sin(x) - (x - x^3/6)$ is increasing on this interval?

• For the function $f(x) = x^5 - x + 1$ identify any intervals where the function is decreasing within $[-5, 5]$.

• Two phone plans are given as follows: Plan 1 is 40 dollars per month for the first 5Gb of data and 2 dollars additional for any additional data usage; Plan 2 is a flat rate of 75 dollars per month, unlimited data. Graphically find the "break-even" point.