Calculus Test 1

1. Specify the domain of the function y = |x – 1|.

2. Which of the following is a true statement about the graph of the equation $y = x^4 +1$?
3. 1. It is symmetric about the x-axis.
2. It is symmetric about the y-axis.
3. It has two x-intercepts.
4. It has no x-intercepts.

a. Only statements 2 and 3 are true.
b. Only statements 2 and 4 are true.
c. Only statements 1 and 3 are true.
d. Only statement 2 is true.

4. Which of the following statements are true of the graph of $y = \frac{2x-1}{x+1}$?

1. It has no x-intercept.
2. It has a slant asymptote y=2x.
3. It has a vertical asymptote at $x=-1$.
4. It has a horizontal asymptote at y=2.

a. Statements 1, 2, and 3 are true.
b. Statements 3 and 4 are true.
c. Statements 2, 3, and 4 are true.
d. All four statements are true.

5. Let $f(x) = 2 \cos x$. The domain of $f^{-1}(x)$ is

1. $[-1,1]$ 2. $(2, \infty)$ 3. $(-\infty, \infty)$ 4. $(-2,2)$ 5. $[-2,2]$

6. Simplify as far as possible. $\ln e +a^{\log_{a}5}- \log100+10^0 - \log_{3} \frac{1}{3}$
7. Solve for x.

a. $\log_3 x + \log_3(2x+5)=1$

b. $\frac{1}{\sqrt{2}}=4^{x}$

8. $f(x)=(x-3)(x+1)^2(x-1)^4$
9. Identify the parts of the following composite function. $f(g(h(j(x))))=\frac{1}{\sqrt{\log(x-1)}}$
10. Consider the picture below. Find the equation of the line L.

11. The following table represents a function of the form $f(x)=ab^{x}$. Find the equation of the function.

$\begin{tabular}{| l | c | r |} \hline x & f(x)\\ \hline 0 & 6\\ \hline 1 & 18\\ \hline 2 & 54\\ \hline 3 & 162\\ \hline 4 & 486\\ \hline \end{tabular}$
12. Given $a^{m}=2$, $a^{n}=3$, $b^{m}=4$, and $b^{n}=5$,
use the properties of exponentials to determine $(a^{3n}b^{m+n})^{\frac{1}{3}}$.

13. Evaluate:
1. $\sin \frac{\pi}{3}$
2. $\tan \frac{3\pi}{4}$
3. $\cos \frac{5\pi}{6}$
14. Find all solutions to the equaion, $\tan^{2}x=\tan x$, such that $x \in [0,2\pi]$.
15. Find $f^{-1}(x)$ if $f(x) = \sqrt{e^{x}+2}$.