10.3 NOTES-ANSWERS.jnt

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

TANGENTS WITH PARAMETRIC EQUATIONS

Sample Problem #1:

Find

dy

dx

for the curve given by

sin( )

x

t

and

cos( ).yt



AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

LET’S DETERMINE

2

dy

dx

FOR THE PARAMETRIC EQUATION ABOVE.

Sample Problem #2

:

For the curve given by

x

t

and



2

1

4

yt

,

0t 

find the slope and concavity at the point





2,3 .

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Sample Problem #3:

Find the tangent line(s) to the parametric curve given by

53

4

x

tt



and

2

yt



at





0,4

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Horizontal tangents will occur where the derivative is zero and that means that we’ll get horizontal

tangent at values of t for which we have

0

dy

dt



.

Vertical tangents will occur where the derivative is not defined and so we’ll get vertical tangents at

values t for which we have

0

dx

dt



Sample Problem #4

:

Determine the x-y coordinates of the points where the following parametric equations will have

horizontal or vertical tangents.

3

() 3

x

tt t

and

2

39yt





AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Sample Problem #5:

Determine the values of t for which the parametric curve given by the following set of parametric

equations is concave up

and concave down.

2

1

x

t

and

75

yt t





AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

AREA WITH PARAMETRIC EQUATIONS

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Sample Problem #6:

Determine the area under the parametric curve given by the following parametric equations.



6sinx









61 cosy







02







AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

ARC LENGTH WITH PARAMETRIC EQUATIONS

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Sample Problem #7

:

Determine the length of the parametric curve given by the following parametric equations.

3sin( )

x

t

3cos( )yt

02







Since this is a circle we could have just used the fact that the length of the circle is just the

circumference of the circle. This is a nice way, in this case, to verify our result.

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Sample Problem #8

:

Determine the length of the parametric curve given by the following parametric equations.

3sin(3 )

x

t

3cos(3 )yt



02







LET’S TAKE A LOOK AT ONE POSSIBLE CONSEQUENCE OF A CURVE

BEING TRACED OUT MORE THAN ONCE, AND US TRYING TO FIND THE

LENGTH OF THE CURVE WITHOUT TAKING THIS INTO CONSIDERATION.

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

SURFACE AREA WITH PARAMETRIC EQUATIONS

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

Sample Problem #9:

Determine the surface area of the solid obtained by rotating the following parametric curve about the

x-axis.

3

cos ( )x





3

sin ( )y





0

2







AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2

HOMEWORK: DAY 1 – ODDS, DAY 2 – EVENS

AP CALC. BC Section 10.3: PARAMETRIC EQUATIONS AND CALCULUS, pg. 719 – DAY 1 AND 2