HW4 Solutions Part 1

Not your typical homework solutions, but it should be useful.

Section 2.7
page 150
3a) There are two parts to 3a. You should be familiar with both definitions of the derivative. Page 146 and page 147. Sometimes one will be easier to work with than the other. For this problem, you can use the point-slope formula to find the equation of a line.

6) Take the derivative (using definition 1 or equation 2, though equation 2 will be easier to work with) and evaluate it at x = -1. This is the slope of the tangent line.

24a) Like number 6, using equation 2 will be easier to work with, but be familiar with both computations, because a test question might require you to use one method or the other.

31) to 36) Identify whether or not the derivative is using definition 1 or equation 2. If using definition, then identify f. Then you can determine a. If using equation 2, then a is given in the denominator (x-a), and f(x) is clearly in the numerator (f(x)-f(a)).

Section 2.8
page 163
19) Read the instructions. On the homework, you might have skipped out on answering a part of the problem and gotten away with it, but if you skip a part of a problem on the test, you won't be happy. Identify all the parts of a question. For this problem we have
i) Find the derivative using the definition of a derivative.
ii) State the domain of the function
iii) State the domain of the derivative.
For this problem the domain of both the function and it's derivative is all real numbers. Do not confuse the domain with the range. Please review the concepts of domain and range on page 11. In this case, the range of f(x) is all real numbers and the range of it's derivative is just the set {1/2}.

If you are asked for the definition of a derivative on an exam, you should write down (2) on page 154.

26) So use the definition to compute the derivative. This problem requires being careful with the algebra. First you need to write f(x+h)-f(x) over a common denominator. Then expand the products. Note: You should get a feeling of when you need to expand products, and when it's okay to leave them alone. When combining fractions, then you'll probably need to expand products.

As a matter of style, once you take the limit, you should stop writing the limit symbol. Technically it's not wrong if you continue writing it, but it's not clear (to me) you know understand the concept. For example, taking the limit as x approaches 0, the following two sequences of equality are both correct, but convey (at least to me) different meanings:
i) lim x+x^2 = lim 0+0 = 0
ii) lim x+x^2 = 0+0 = 0.
I don't know if anybody has any input on this.

54) I don't have much to say about this problem itself, but I do want to mention that when computing the limit of a function (page 88), you don't have to compute the left and right limits separately unless the function is a piecewise function.

Chapter 2 Review
page 167
3) Composition of continuous functions
4, 5, 6, 8, 11) factor the polynomials and cancel common factors
7) Expand and then cancel the h in the numerator and denominator
9) page 94, limits going to infinity
10) from v greater than 4, 4-v is negative, so the denominator |4-v| = -(4-v) = v-4.