# 1.5: Exercises

• • Contributed by Marcia Levitus

To see if you are on track, solve the following exercises using only the formula sheet (no calculators, computers, books, etc!).

1. Draw the straight line that has a $$y$$-intercept of 3/2 and a slope of 1/2.
2. Express $$\frac{3}{4} - \frac{2}{3} + 1$$ as a single fraction.
3. Simplify $$(a -4a^3)/a^{-2}$$.
4. Express $$\ln 8 − 5 \ln 2$$ as the logarithm of a single number.
5. Given $$\ln P = − \frac{a}{RT} + b \ln T + c$$, where $$a$$, $$b$$, $$c$$ and $$R$$ are constants, obtain $$\frac{d(\ln P)}{dT}$$
6. Obtain $$\frac{dy}{dx}$$
1. $$y = \sin xe^{mx}$$ (m is a constant).
2. $$y = \frac{1}{\sqrt{1−x^2}}$$
7. Obtain the first, second and third derivatives of
1. $$y = e^{−2x}$$
2. $$y = \cos(2x)$$
3. $$y = 3 + 2x − 4x^2$$
8. Evaluate $$\int_0^{\pi} \cos 3 \theta d \theta$$.
9. Use the properties of integrals and your previous result to evaluate $$\int_{\pi}^0 \cos 3 \theta d \theta$$. What about $$\int_0^{\pi/4} \cos 3 \theta d \theta + \int_{\pi/4}^{\pi} \cos 3 \theta d \theta$$?
10. Given $$f(x) = \left\{\begin{matrix} 0 & \text{if } x<0 \\ 3+2x & \text{if } 0 <x<1 \\ 0 & \text{if } x>1 \end{matrix}\right.$$ Sketch $$f(x)$$ and calculate $$\int_{- \infty}^{\infty} f(x) dx$$
11. What is the value of this integral? $$\int_{- \infty}^{\infty} xe^{-x^2} dx$$
12. Sketch $$\sin(x/2)$$. What is the period of the function?
13. The plots below (Figure $$\PageIndex{1}$$) represent the following functions: $$y = 3e^{−x/2}, ~ y = 3e^{−x}, ~ y = 3e^{−2x}$$ and $$y = 2e^{−2x}$$. Which one is which?