MTH132 Section 5 & 18, Quiz 1



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MTH132 Section 1 & 12, Quiz 3

Feb 20, 2009 Instructor: Dr. W. Wu

Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each question. The total time allowed for this quiz is 15 minutes.

1 [3 pts each]. Find [pic] for following functions (Do Not Simplify)

(a) [pic]

[pic] （apply power chain rule）

So [pic]

(b) [pic] (chain rule)

[pic]

(c) [pic]

[pic] (use chain rule, the inside function of the first term is[pic], and the inside function of the second term is[pic]).

So [pic]

(d) [pic] (use implicit differentiation)

consider[pic]as a function of[pic] (implicitly)

apply[pic]to this implicit equation: [pic]

so [pic]

2. [4 pts]. Show that [pic] lies on the curve [pic]. Then find an equation for the tangent line to the curve at this point.

Substitute [pic] into the left hand side [pic]. So it is on the curve.

Consider[pic]as a function of[pic] (implicitly), and use implicit differentiation

[pic]

So [pic]. At [pic], the slope is [pic]

Use point-slope equation: the tangent line at[pic] is [pic]

3. Suppose the parametric equations of a curve are [pic], [pic].

(a) [2 pts]. Find[pic]by using parametric formula.

[pic]

(b) [2 pts]. Let[pic]be the point when [pic]. Find the slope of the tangent line to this curve at[pic].

The slope is the derivative at [pic], so it is [pic]

(c) [bonus 2 pts]. Find a Cartesian equation for this curve, then find[pic]at[pic]by using implicit differentiation.

[pic], [pic], by trigonometric identity [pic]

A Cartesian equation for this ellipse is [pic]. (Ex 3.5 #69)

Using implicit differentiation: [pic][pic]. When[pic], [pic],[pic]. So the slope is [pic].

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