Math 215 Practice set 1 (due 1/10/06)

Read section 7.1

Remember your calculus? Differentiate the following functions:

Practice 1   $x^5+3x+2$

Practice 2   $2^x$

Practice 3   $\ln x$

Practice 4   $x^5e^x$

Practice 5   $\sqrt{\ln x}$

Practice 6   $x/\ln x$

In Brief Calculus and its applications, do the following problems in section 7.1:

7.1 #1, 3, 5, 7

Math 215 To-turn-in set 1 (due 1/10/06)

Remember your calculus? Differentiate the following functions:

Problem 1   $x^4+3x^2$

Problem 2   $3e^x$

Problem 3   $x\ln x$

Problem 4   $\sqrt{5+3x^2}$

Problem 5   $\frac{x^2}{\ln x}$

Problem 6   $\sqrt{x}e^{3x}$

In Brief Calculus and its applications, do the following problems in section 7.1:

To turn in:

7.1 #2, 4, 6, 8

Math 215 Practice set answers

1.

$$\begin{eqnarray*} f(x)&=&x^5+3x+2\\ f'(x)&=&5x^4+3 \end{eqnarray*}$$

2.

$$\begin{eqnarray*} f(x)&=&2^x\\ f'(x)&=&(\ln 2)2^x \end{eqnarray*}$$

3.

$$\begin{eqnarray*} f(x)&=&\ln x\\ f'(x)&=&1/x \end{eqnarray*}$$

4.

$$\begin{eqnarray*} f(x)&=&x^5e^x\\ f'(x)&=&x^5\frac{d}{dx}(e^x)+\frac{d}{dx}(x^5)e^x\\ &=&x^5e^x+5x^4e^x \end{eqnarray*}$$

5.

$$\begin{eqnarray*} f(x)&=&\sqrt{\ln x}\\ &=&(\ln x)^{1/2}\\ f'(x)&=&\frac{1}{2}... ...2}\frac{d}{dx}(\ln x)\\ &=&\frac{1}{2}(\ln x)^{-1/2}\frac{1}{x} \end{eqnarray*}$$

6.

$$\begin{eqnarray*} f(x)&=& x/\ln x\\ f'(x)&=&\frac{\ln x\frac{d}{dx}(x)-x\frac{d... ...n x)(1)-x\frac{1}{x}}{(\ln x)^2}\\ &=&\frac{\ln x-1}{(\ln x)^2} \end{eqnarray*}$$

or equivalently, you could do:

$$\begin{eqnarray*} f(x)&=& x/\ln x\\ &=&x(\ln x)^{-1}\\ f'(x)&=&x\frac{d}{dx}(\... ... x)^2}+\frac{\ln x}{(\ln x)^2}\\ &=&\frac{\ln x - 1}{(\ln x)^2} \end{eqnarray*}$$

Answers to book problems found in the back of the book.