Sample Problems, 3.2

Sample Problems, Lesson 3.2

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3.2, #16, Given Problem.

Differentiate:	3.2, #16, Given Problem.
	First, we multiply out the quantity before differentiating.
	Here we differentiate using the power rule on the first term, and the leave the derivative of the second term until the next step.
	Here we use the product rule on the remaining derivative.

3.2, #24, Given Problem.

	3.2, #24, Given Problem.
	Part (a). Here, we are using the quotient rule.
	Multiply out and simplify the numerator. Then factor out a -1. This allows a cancellation with one factor in the denominator.
	Here we see the original function (green) and our derivative. Does the yellow appear to be consistent with being the derivative of the green curve? It does because, 1) the green curve is always decreasing and this agrees with the yellow curve being always negative, 2) on the "middle branch" of the green curve we note that it has a negative slope, which at first increases, gets to as low as about -1 and then decreases again. The yellow curve is consistent with this. 3) the first and third branches of the green curve are also consistent with the yellow derivative curve.

3.2, #40, Given Problem.

	3.2, #40, Given Problem.
	Part (a). We take the given function and differentiate using the product rule.
	We now differentiate a second time. There are two separate product rules to use here.
	We simplify and rearrange terms, and we have shown what was to be shown.
	Part (b). Here we again differentiate. This time we have three terms to deal with.
	Here we differentiate a fourth time. Now we have four terms to deal with.
	Here we observe the pattern which was developing in our first four derivatives. We noticed that when writing the terms in descending order, the coefficients are the same coefficients as the binomial theorem, hence a portion of this formula is from the binomial theorem.

	*Assignment*
1-35. Odds
Lesson 3.2, Pages 204-205