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# 1.5: Exercises

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?
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