Chapter 3 sections 3.3 and 3.4 What is differentiation and what are the methods of differentiation?

Section 3.3 Rules of Differentiation day 1 What is the power rule for derivatives?

Power Rules  Find the derivative of each: f(x)f(x) = x 2 f(x) = x 3 Do you see a pattern?

explore  Does the pattern hold now? f(x) = 2xf(x) = 3x 2 What modification has to be made?h

Derivative of a Constant  Find the derivative f(x) = 2f(x) = 3 Would it help to think of these like this? f(x) = 2x 0 f(x) = 3x 0 Find the derivative: consider it in graphical form. What is the Slope of the line represented by A constant?

Sum & Difference Differentiation (differentiation of a polynomial)  Find the derivative: f(x) = 3x 2 + 2x  (hint: The limit of a sum = the sum of the limits— take the derivative of each part) f(x) = 3x 2 + 2x

Product Rule  Find the derivative of the product If g(x) = x + 2 and h(x) = 3x f(x) = (x + 2)(3x 2 + 1) f(x) = 3x 3 + 6x 2 + x + 2 f‘(x) = 9x x + 1  Do you really want to multiply these to get the solution? If r(x) = x 3 + 2x 2 + 3x + 5 And s(x) = 4x 5 – 2x 3 +3x 2 – 6x + 1  Find g’ and h’ g‘ (x) = 1 and h’(x) = 6x TRY: f’(x) = g(x)h’(x) + h(x)g’(x) f‘(x) = (x+2)(6x) + (3x 2 + 1)(1) = 6x x + 3x = 9x x + 1 Rule=1 st d(2 nd ) + 2 nd d(1 st )

Quotient Rule:

Consider

 f(x) = 3x 3 – 4x 2 + 2x – 5  Find f”(x)

Homework Page(s): – 20b

Section 3.3 day 2 What is the power rule for derivatives?

Homework Page(s): , 26, 28, 30

Section 3.3 day 3 What is the power rule for derivatives?

IN CLASS Page(s): , 34, 35

Section 3.4 Velocity & rates of change day 1 How is the rate of change related to the derivative?

Rate of Change  Rate of Change is equivalent to instantaneous rate of change  Therefore the derivative of a function

Example 1  Find the rate of change of the area of a circle with respect to its radius  A= πr 2  A’ = 2πr  When r = 5  A’ = 10π

Relationship between distance velocity and acceleration  Motion  Distance (s) = f(x)  Velocity (v) = f’(x)  Acceleration (a) = f”(x)

Example  A projectile is hurled upwards and follows a path defined by s= t + 16t 2  Find the max height of the object S = – 16t t + 6 V = -32t + 64 A = -32 What do we know about the graph? 2 nd power indicates parabolic flight Neg on the leading coefficient indicates a max pt. (max and mins occur when the tangent is horizontal, so the slope of the tangent is 0)

Example S = – 16t t + 6 V = -32t + 64 A = = -32t = -32t 2 = t The max occurs when t = 2 at S = – 16(2) (2) – 6 = 72 ft.

Example S = – 16t t + 6 V = -32t + 64 A = -32 How fast is it traveling when it is 54’ high? 54 = – 16t t = – 16t t – 48 t = 3 or 1 V(1) = 32 ft/sec v(3) = -32 ft/sec

Example S = – 16t t + 6 V = -32t + 64 A = -32 What is the acceleration? A = -32 ft/sec 2 (it starts to slow down as soon as it leaves the ground)

Example S = – 16t t + 6 V = -32t + 64 A = -32 When does it hit the ground? 0 = – 16t t + 6 Use the quad formula T = 4.09 sec or -.09 sec What does this tell us? The object did not start on the ground Where did it start? S = – 16(0) (0) + 6 = 6’

Uses of derivatives  Economics  Cost equations =c(x)  Marginal cost = c’(x)  Revenue = r(x)  Marginal revenue = r’(x)

Homework Page(s): 129 1, 3, 5, 9, 13

Section 3.4 day 2 How is the rate of change related to the derivative?

Homework Page(s): Review