Finding the Derivative/Rate of Change.  The derivative of a constant is 0. That is, if c is a real number, then 1. Sketch a graph to demonstrate this.

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Presentation transcript:

Finding the Derivative/Rate of Change

 The derivative of a constant is 0. That is, if c is a real number, then 1. Sketch a graph to demonstrate this rule. 2. Prove that the derivative of a constant, c, is 0.

A. y = 7 B. f(x) = 0 C. s(t) = -3 D. y = kπ² A. y’ = 0 B. f’(x) = 0 C. s’(t) = 0 D. y’ = 0

 If n is a rational number, then the function is differentiable and

A. f(x) = x³ B. g(x) = C. y = g’(x) = f’(x) = 3x² y’ = -2/x³

 If f is differentiable and c is a real number, then c·f(x) is also differentiable and  Prove the Constant Multiple Rule.

 f(x) = 2x²  g(x) = -4x³  y = (¾)x³

FUNCTIONREWRITEDERIVATIVESIMPLIFY

 Pg 113 #1-9 odds, 25–30 odds

 Constant Rule  Power Rule  Constant Multiple Rule

 The sum or difference of two differentiable functions is differentiable and is the sum or difference of their derivatives.  Prove the Sum Rule

 f(x) = x³ - 4x + 5  g(x) =

 Find the equation of the line tangent to the graph f(x) = x² when x = -2.

 Pg 113 #11-18, 31-34, 53, 55

1.No Calculator 2. 3.

 Find the derivative. 1. y = 2sin x 2. y = 3. f(x) = x + cos x

 Position Function – A function, s(t), that gives the position of an object as a function of time.  So  Therefore, POSITION → VELOCITY

 If a billiard ball is dropped from a height of 100 ft, then its height s at time t is given by the position function, s(t) = -16t² where s is in feet and t is in seconds. Find the average velocity for the following time intervals. A. [1,2] B. [1, 1.5] C. [1,1.1]

 PG 115 #21-24, 81-86, *Just find avg. velocity on 87-90

 The position of any free falling object can be given by the position function Where is initial velocity, is initial height, and g is gravity.  Velocity at a particular instant is

 Velocity can be negative, 0, or positive.  Speed is the ___________ ___________ of velocity.

 At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s(t) = -16t² + 16t + 32 where s is in feet and t is in seconds. A. When does the diver hit the water? B. What is the diver’s velocity at impact? C. How fast was the diver going?

 Pg 115 #87-90 *find instantaneous velocity at each endpoint and compare to previous night’s answers, 91-94