AP CALCULUS

Chen DongyangDate:12/1/2007

Teacher: Dela Cruz

Motion Like Expressions, Motion Math is a simple program

ming environment within After Effects. Motion Math scripts can create key frames for your layer based on values from another layer or property. For instance, you could use Motion Math to have one layer mimic another’s position changes. Unfortunately, ‘relationships’ created via Motion Math are temporary, and only reflect the conditions at the time the script is executed. Afterwards, changes to one layer will not be reflected in the other layer, unless the script is re-applied.

NO.1

A circle is increasing in area at the rate of .How fast is the radius increasing when the radio is 2 in?

2

16in

S

Answer Step 1: Use the expression that relates the area of

a circle to its radius: A= . Step 2 : Take the derivative of the expression with

respect to t:

Step 3:Plug In and r=2

Step 4: Last you can solve for , you will get

2r

2dA dr

rdt dt

16dA

dt

16 2 (2)dr

dt

dr

dt

4sec

dr in

dt

No.2

A spherical balloon is expanding at a rate of . How fast is the surface area of the balloon expanding when the radius of the balloon is 4 in?

3

60sec

in

Step 1: You are given the rate at which the volume's expanding, and you know the equation that relates volume to radius. But you have to radius area to surface area as well, because you've got to find the surface area's rate of change. This means that you'll need the equation for volume and surface area of sphere:

24A r

34

3v r

You`re trying to find ,but A is given in terms of r,

so you have to get first. We can get from equation of volume:

And ,so

24dV dr

rdt dt

dA

dt

dr

dt

dr

dt

60dV

dt 15

16 sec

dr in

dt

Step 2:Now ,we take the derivative of the other equation with respect to t:

We can plug in for r and from the previous step and we get:

One final example.

8dA dr

dt dt

2 215 4808 (4) 30

16 16 sec sec

dA in in

dt

No.3

A 25-foot long ladder is leaning against a wall and sliding away from the base of the wall at the a rate of 15 feet/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 7feet from ground?

Step 1: we can know the ladder forms a right triangle with the wall. We let x stands for distance from the foot of the ladder to the base of the wall. We let y represent the distance from the top of the ladder to the ground. We can get a equation:

Step 2:We take the derivative of the equation :

2 2 25x y

2 2 0dx dyx ydt dt

Step 3: We know the rate that the ladder is sliding away from the wall:

We also are given the distance from the ladder to the top of the wall (y) is 7feet. We can get x:

15dx

dt

2 225 7x

24x feet

Step 4: We can put the all information into the derivative:

2(24)(15) 2(7) 0dy

dt

360

7 sec

dy feet

dt

Above the three examples are from”Cracking the AP CALCULUS AB & BC Exams”.

Thank you for reading.