HW4 Solutions Part 2

Again, not your typical homework solutions, but what's written here should be useful.

I last left off classifying my thoughts about finding the limit to problems 3-20 for the Chapter 2 Review on page 167.

Notation:
\sqrt{f} means the square root of f
\frac{f}{g} means the quotient f/g
a^b means a to the power b

Chapter 2 Review
page 167
12, 17) So there's the square root of one function f minus another function g,
\sqrt{f} - g then multiply and divide by the conjugate \sqrt{f} + g
If it was \sqrt{f} + g then multiply and divide by the conjugate \sqrt{f} - g.
Note: when you multiply and divide by the conjugate, you should be leaving the "good" part alone, don't multiply things together.
For example: \frac{\sqrt{f}-g}{h}*\frac{\sqrt{f}+g}{\sqrt{f}+g}=\frac{f-g^2}{h(\sqrt{f}+g)}

Note: the reason you multiply and divide by the conjugate is to eliminate some part of the function which is behaving badly.

13, 14, 16) Let d be the largest degree occurring in the denominator and numerator. Multiply and divide by \frac{1}{x^d}. If p(x) is a polynomial of degree 2n, then you can think of \sqrt{p(x)} as having degree n, for the purposes of figuring out the largest degree. So for problems 13, 14, and 16, d=1, d=1, and d=4, respectively.

20) Combine fractions.

34) Let f(x)=exp{-x^2}-x, it is continuous at all real numbers. In particular it is continuous on the interval [0,1]. Then apply the intermediate value theorem with N=0, see page 126. Then there exists a number c in (0,1) such that f(c)=0. Then exp{-c^2}=c.

Section 3.1
Note: obviously you don't have to state all the rules, or do all the steps one-by-one, but I describe all the rules that are implicitly being used.
4) Derivative of a constant funciton
7) Sum rule, constant multiple rule, power rule and derivative of a constant function
9) constant rule, power rule. Answer: t^3.
15) constant rule, Power rule
22) because you don't have the product rule in this section, you would expand the product and then use the power rule.
\frac{d}{dx}\left(\sqrt{x}(x-1)\right)=\frac{d}{dx}\left(\sqrt{x}x-\sqrt{x}\right)=\frac{d}{dx}\left(x^{\frac{3}{2}}-x^{\frac{1}{2}}\right)=\frac{3}{2}x^{\frac{1}{2}}-\frac{1}{2}x^{-\frac{1}{2}}

25) Derivative of a constant function
30) Like 22, because you don't have the product rule yet, Expand and power rule

I recommend doing problems that weren't assigned between 3 and 32. Let me know in the comments if you have any questions.

36) Take derivative after expanding and using the power rule to find the slope at x=1. Then find the equation of the tangent line.

51) Take the derivative of the function and set it equal to 0. Either factor or use the quadratic formula to find the zeroes of the derivative. The problem is asking for the points on the curve so your answers should be of the form (x,y). In particular (x,f(x)), where x is a root of the derivative.

61) Be able to do this! This particular problem or a problem like it, is a simple, yet conceptual problem. I would start by writing down the definition of the derivative. Then


65) I have shown different ways to do this problem. Let f(x)=ax^3+bx^2+cx+d. Then f'(x)=3ax^2+2bx+c. Then we have f(-2)=6, f(2)=0, f'(-2)=0, and f'(2)=0. Thus we have four (linear) equations for the four variables a, b, c, and d. Use algebra to solve for a, b, c, and d.