CALCULUS BC



CALCULUS BC

WORKSHEET ON PARAMETRICS AND CALCULUS

Work these on notebook paper. Do not use your calculator.

On problems 1 – 5, find[pic].

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

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6. A curve C is defined by the parametric equations[pic].

(a) Find [pic] in terms of t.

(b) Find an equation of the tangent line to C at the point where t = 2.

7. A curve C is defined by the parametric equations[pic].

(a) Find [pic] in terms of t.

(b) Find an equation of the tangent line to C at the point where t =[pic].

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On problems 8 – 10, find:

(a) [pic]in terms of t.

(b) all points of horizontal and vertical tangency

8. [pic]

9. [pic]

10. [pic] ______________________________________________________________________________

On problems 11 - 12, a curve C is defined by the parametric equations given. For each problem, write an integral expression that represents the length of the arc of the curve over the given interval.

11. [pic]

12. [pic]

Answers to Worksheet on Parametrics and Calculus

[pic]

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[pic]

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[pic]

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[pic]

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[pic]

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6. (a) [pic]

(b) When t = 1, [pic]so the tangent line equation is

[pic]

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7. (a) [pic]

(b) When [pic] [pic] so the tangent line equation is

[pic]

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8. (a) [pic]

(b) A horizontal tangent occurs when[pic] so a horizontal tangent

occurs when [pic] which is at t= 2. When t = 2, x = 7 and y = [pic]4 so a horizontal

tangent occurs at the point[pic]. A vertical tangent occurs when [pic] Since[pic], there is no point of vertical tangency on this curve.

9. (a) [pic]

(b) A horizontal tangent occurs when [pic] so a horizontal tangent occurs when [pic] which is at[pic]. When t = 1 , x = 1 and y = [pic] 2, and when

[pic], x = 3 and y = 2 so a horizontal tangent occurs at the points [pic]

A vertical tangent occurs when [pic] so a vertical tangent occurs when

[pic]. When[pic],[pic] so a vertical tangent occurs at the

point[pic].

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10. (a) [pic]

(b) A horizontal tangent occurs when [pic] so a horizontal tangent

occurs when [pic] which is at[pic]. When t = [pic] , x = 3 and y = 3, and when

[pic], x = 3 and y = [pic]5 so a horizontal tangent occurs at the points[pic].

A vertical tangent occurs when [pic] so a vertical tangent occurs when

[pic]. When[pic][pic]and when [pic]

so a vertical tangent occurs at the points[pic].

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11. [pic]

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12. [pic]

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