3.2 Product and Quotient Rules

Mathematica script by Chris Parrish,
cparrish@sewanee.edu

Problems from James Stewart,
"Calculus," Second Edition, Brooks/Cole, 2001

Mathematica knows the Product and Quotient Rules.

Stewart, section 3.2

"D" is the differentiation operator.

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Well, that version of the quotient rule may not be what we expected.

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That looks more familiar.

Calculate a Derivative.

Stewart, cf. exercise 3.2.15

Use Mathematica to define a certain function y[v] and to calculate its derivative.

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Find a Tangent Line.

Stewart, cf. exercise 3.2.20

Let's use Mathematica to define and graph a certain function y[x]
and then to find the equation of the tangent line to the graph of y at x = 2.

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Now, calculate the derivative of y.

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Find the equation of the tangent line to the graph of y at x = 2.

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Let's check that result by plotting y and its tangent line on the same screen.

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How Many Tangent Lines to the Graph of y(x) Pass through P(1,2)?

Stewart, cf. exercise 3.2.37

How many tangent lines to the graph of y[x] pass through the point P(1,2) ?

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From the picture, it seems evident that there should be two, corresponding to the two branches of the graph.
To be tangent to the graph of y[x], a line would have to have an equation of the form tan[x] = y'[a] (x - a) + y[a]
for some appropriate value of a.

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Moreover,the tangent line must go through the point P(1,2), so that 2 = tan[1].
Let's set up that relationship.

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Those are the two appropriate values of a.
Let's use them to calculate the two tangent lines.

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One tangent line looks pretty convincing, and the other is at least plausible.
Extend the domain of x to the left to get a better image of the second line.

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The leftmost point of tangency is at x = a1.

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Let's focus on that point.

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Yup! Pretty convincing.

Created by Mathematica  (April 19, 2004)