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Parametric Equations

The document discusses parametric equations and their derivatives in calculus, including how to find the first and second derivatives of a curve defined by parametric equations. It provides several exercises for calculating derivatives, finding tangent lines, and setting up integral expressions for arc lengths. The exercises are designed for practice without the use of calculators.

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H202251
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21 views2 pages

Parametric Equations

The document discusses parametric equations and their derivatives in calculus, including how to find the first and second derivatives of a curve defined by parametric equations. It provides several exercises for calculating derivatives, finding tangent lines, and setting up integral expressions for arc lengths. The exercises are designed for practice without the use of calculators.

Uploaded by

H202251
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Parametric Equations and Calculus


If a smooth curve C is given by the equations x  f t and y  g t , 
dy
 
then the slope of C at the point x, y is given by
dy dt

dx dx
where
dx
dt
 0,

dt
d  dy 
 
d y d  dy  dt  dx 
2
and the second derivative is given by     dx .
dx 2 dx  dx 
dt
Ex. 1 (Noncalculator)
dy d2 y
Given the parametric equations x  2 t and y  3t  2t , find
2
and .
dx dx 2

__________________________________________________________________________________
Ex. 2 (Noncalculator)
Given the parametric equations x  4cost and y  3sint , write an equation of the tangent line to the
3
curve at the point where t  .
4

__________________________________________________________________________________
Ex 3 (Noncalculator)
Find all points of horizontal and vertical tangency given the parametric equations
x  t 2  t, y  t 2  3t  5.

__________________________________________________________________________________
Ex. 4 (Noncalculator)
Set up an integral expression for the arc length of the curve given by the parametric
equations x  t 2  1, y  4t 3  1, 0  t  1. Do not evaluate.
CALCULUS BC
WORKSHEET ON PARAMETRICS AND CALCULUS

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


dy d2 y
On problems 1 – 5, find and .
dx dx 2
1. x  t 2 , y  t 2  6t  5
2. x  t 2  1, y  2t 3  t 2
3. x  t , y  3t 2  2t
4. x  lnt, y  t 2  t
5. x  3sint  2, y  4cost  1
_____________________________________________________________________________
6. A curve C is defined by the parametric equations x  t 2  t  1, y  t 3  t 2 .
dy
(a) Find in terms of t.
dx
(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 x  2cost, y  3sint .


dy
(a) Find in terms of t.
dx

(b) Find an equation of the tangent line to C at the point where t =
.
4
______________________________________________________________________________
On problems 8 – 10, find:
dy
(a) in terms of t.
dx
(b) all points of horizontal and vertical tangency
8. x  t  5, y  t 2  4t
9. x  t 2  t  1, y  t 3  3t
10. x  3 2cost, y  1 4sint
______________________________________________________________________________
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. x  t 2 , y  t 3 , 0  t  2
12. x  e2t  1, y  3t  1,  2  t  2

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