BC Calculus Assignment 35: Exponentials Worksheet A

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

I. Express the logarithms in terms of $$\ln{ 2}$$ and $$\ln {3}$$ using the rules of arithmetic for logarithms. For example, $$\ln {1.5} = \ln { \left( \frac{3}{2} \right) } = \ln {3} - \ln {2}$$.
1. $$\ln { \left( \frac{4}{9} \right) }$$
2. Solution $$2\ln{2} - 2\ln{3}$$
3. $$\ln{12}$$
4. Solution $$2\ln{2} + \ln{3}$$
5. $$\ln{\sqrt[3]{9}}$$
6. Solution $$\frac{2}{3}\ln{3}$$
7. $$\ln{(3 \sqrt{2})}$$
8. Solution $$\ln{3} + \frac{1}{2}\ln{2}$$
II. Find $$\frac{dy}{dx}$$
1. $$\ln \left( x^2 \right)$$
2. Solution $$\frac{2}{x}$$
3. $$\ln { \left( \frac{1}{x} \right) }$$
4. Solution $$-\frac{1}{x}$$
5. $$y = \ln {(2 - \cos{x})}$$
6. Solution $$\frac{\sin{x}}{2 - \cos{x}}$$
7. $$y =\ln {\left( \ln {x} \right)}$$
8. Solution $$\frac{1}{x \ln {x}}$$
9. $$y = x \ln{x} - x$$
10. Solution $$\ln{x}$$
11. $$y = \frac{x \sin{x}}{\sqrt{\sec{x}}}$$
12. Solution $$\ln{y} = \ln{x} + \ln{ ( \sin {x} ) } - \frac{1}{2} \ln{\sec {x}}, so \frac{y'}{y} = \frac{1}{x} + \frac{\cos{x}}{\sin{x}} - \frac{1}{2} \frac{\sec {x}\tan {x} } { \sec{x}}$$. $$y' = y \left( \frac{1}{x} + \cot {x} - \frac{1}{2} \tan {x} \right)$$, and substitute back for $$y$$.
13. $$y = \sqrt{\frac{x}{(x + 1)(x - 3)}}$$
14. Solution $$\ln{y} = \frac{1}{2} \left( \ln{x} - \ln{(x + 1)} - \ln{(x - 3)} \right)$$, so $$\frac{y'}{y} = \frac{1}{2} \left( \frac{1}{x} - \frac{1}{x + 1} - \frac{1}{x - 3} \right)$$. Then $$y' = \frac{1}{2} y \left( \frac{1}{x} - \frac{1}{x + 1} - \frac{1}{x - 3} \right)$$ and substitute back for $$y$$.
III. Evaluate the following integrals analytically. Support with your calculator.
1. $$\int_{2}^{5} \frac{dx}{x} =$$
2. Solution $$\ln{5} - \ln{2}$$
3. $$\int_{3}^{4} \frac{dx}{2x} =$$
4. Solution $$\frac{1}{2}(\ln{4} - \ln{3})$$
5. $$\int_{1}^{e} \frac{dx}{x - 4} =$$
6. Solution $$\ln{|e - 4|} - \ln {|-3|}$$
7. $$\int_{-1}^{0} \frac{3dx}{x + 3} =$$
8. Solution $$3(\ln{|3|}-\ln{|2|})$$
9. $$\int_{-10}^{-8} \frac{3dx}{x - 5} =$$
10. Solution $$3(\ln{|-13|} - \ln{|-15|})$$