MAT100 Final Exam Questions, Fall 2016

Exam time: 180 minutes.

Work must be done without the aid of a book, class notes, calculator, cell phone, headphones or any other electronic device.

1) Evaluate each of the following:

• $\log_5\frac{1}{125}$,
• $\ln(e^3 \cdot \sqrt{e})$,
• $\sin\left(\frac{4\pi}{3}\right)$,
• $\arccos\left(\frac{\sqrt{3}}{2}\right)$,
• $\sin(\arctan(3))$,
• $\cos(\pi - t)$ if $t\in \left(0,\frac{\pi}{2}\right)$ and $\cos(t) = \frac{1}{5}$,
• $\sin(2 t)$ if $t \in \left(0,\frac{\pi}{2}\right)$ and $\sin(t) = \frac{1}{3}$.

2) Sketch the graph of $f(x) = - x^2 + 6 x - 7$. Clearly mark the vertex and $x$-intercepts.

3) For $\displaystyle f(x) = \frac{12 - 3x}{x+2}$:

• Sketch the graph $y=f(x)$. Clearly mark all asymptotes and intercepts.
• Find the formula for the function $f^{-1}$.
• Find the domain and range of $f^{-1}$.

4) For $f(x) = -2^{x+3} + 4$:

• Explain how the graph of $f(x)$ can be obtained from the graph of the function $g(x) = 2^x$. Clearly mark the order of operations.
• Sketch the graph of $f(x)$. Clearly mark all asymptotes and intercepts.
• Find algebraically the formula for the function $f^{-1}$.
• Sketch the graph of $f^{-1}$. Clearly mark all asymptotes and intercepts.

5) For $g(x) = \tan\left(x + \frac{\pi}{4}\right)$:

• Sketch the graph of $g(x)$ on the interval $[-2\pi, 2\pi]$. Clearly mark all asymptotes and intercepts.
• Determine if $f$ is even/odd/neither.
• Determine if $f$ is periodic and if so, find the period.
• For what $x$ in $[-2\pi,2\pi]$ is $\displaystyle \left| \tan\left(x + \frac{\pi}{4}\right)\right| \leq 1$.

6) Suppose a radioactive isotope is such that at 1pm its mass was 8 grams and at 3pm its mass was 0.125 grams. What was the mass of the isotope at 1:10pm? Simplify your answer.

7) Solve for $x$: $\displaystyle \log_2(x+2) + \log_2(x-1) = 1$.