# 1.5: Exercises

- Page ID
- 106802

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

- Draw the straight line that has a \(y\)-intercept of 3/2 and a slope of 1/2.
- Express \(\frac{3}{4} - \frac{2}{3} + 1\) as a single fraction.
- Simplify \((a -4a^3)/a^{-2}\).
- Express \(\ln 8 − 5 \ln 2\) as the logarithm of a single number.
- Given \(\ln P = − \frac{a}{RT} + b \ln T + c\), where \(a\), \(b\), \(c\) and \(R\) are constants, obtain \(\frac{d(\ln P)}{dT}\)
- Obtain \(\frac{dy}{dx}\)
- \(y = \sin xe^{mx}\) (m is a constant).
- \(y = \frac{1}{\sqrt{1−x^2}}\)

- Obtain the first, second and third derivatives of
- \(y = e^{−2x}\)
- \(y = \cos(2x)\)
- \(y = 3 + 2x − 4x^2\)

- Evaluate \(\int_0^{\pi} \cos 3 \theta d \theta\).
- Use the properties of integrals and your previous result to evaluate \(\int_{\pi}^0 \cos 3 \theta d \theta\). What about \(\int_0^{\pi/4} \cos 3 \theta d \theta + \int_{\pi/4}^{\pi} \cos 3 \theta d \theta\)?
- 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 \(\int_{- \infty}^{\infty} f(x) dx\)
- What is the value of this integral? \(\int_{- \infty}^{\infty} xe^{-x^2} dx\)
- Sketch \(\sin(x/2)\). What is the period of the function?
- 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?