1008 B : Connections

AP CALCULUS

What is the Derivative?

The Derivative represents _____________________________

• A) IN MATH

IN MATH

Find the equation of the line tangent to the function at the point.

EX:3 23 5 2 1y x x x

DEFN: The NORMAL to a curve is the line perpendicular to the tangent at a pointFind the Normal to the curve above.

at x = 2

IN MATH

Determine the point(s) –if any – at which the given function has a HORIZONTAL tangent.

EX:3y x x

Example: Positive Integer Powers, Multiples, Sums, and Differences

4 2Does the curve 8 2 have any horizontal tangents?If so, where do they occur?Verify you result by graphing the function.

y x x

What is the Derivative?

The Derivative represents _____________________________

B) IN PHYSICS

IN PHYSICS

At time t = 0 , Ms. Mangum jumps from a diving board that is 32 feet above the water. the position function is given by

Where s is measured in feet and t is measured in seconds.

a) Find velocity function.

Find her velocity at t = 1/2.

2( ) 16 16 32s t t t

b) Find her velocity when she hits the water.

IN PHYSICS : Particle MotionExample 4: Vertical Motion

A dynamite blast propels a heavy rock straight up with a launch velocity of 192 ft/sec ( about 130.91 mph). Its height is modeled by

A). How high does the rock go?

B). What is the velocity function and the velocity when it reaches 512 ft. above the ground - going up

- coming down.

C). What is the acceleration function?

D). When does the rock hit the ground? What is its velocity?

2( ) 192 16s t t t

IN PHYSICS: Particle Motion

Example 5: Horizontal Motion

A particle moves along line so that its position at any time t > 0 is given by , where x is measured in meters and t in seconds.

2( ) 6 2x t t t

A). Find the displacement of the particle after the first 3 seconds.

B). Find the average velocity of the particle during the first 5 seconds.

C). Find the instantaneous velocity of the particle at t = 6.

D). Find the acceleration of the particle at t = 6.

E). Describe the motion of the particle. At what values of t does the particle change directions?

What is the Derivative?

The Derivative represents _____________________________

C) IN ECONOMICS

IN ECONOMICS

The annual inventory cost C(x) for a certain manufacturer is

Where x is the order size when the inventory is replenished.

a) Find the marginal cost function and the marginal cost when

x = 350.

b) Find the actual change in cost when x is changed from 350 to 351.

1,008,000( ) 6.3C x xx

Last Update

• 09/24/10