Winter wk 3 – Tues.18.Jan.05 Happy MLK day Review: –Derivative = slope = rate of change –Recall polynomial rule for derivatives –Compare to polynomial rule for integrals –Second derivatives Differentiating exponential functions Energy Systems, EJZ

Derivative as slope AB C Which functions have the greatest slope? (b) (a) (b) (a) (b)

Derivative as slope b=3+f a=f b=3f a=f b=3f a=f

Differentiating polynomials

Differentiating & integrating polynomials Differentiation rule: (from last quarter) Integration rule: (from last week) You can check your integration results by differentiating!

Practice – Ch.3.1 Spend 5-10 minutes doing odd numbered problems from p.111 Let’s discuss the physics in #58 and #60.

Higher order derivatives Second derivative = rate of change of first derivative

Polynomials of degree n Polynomial of degree 0: f(x) = a (x 0 ) = a (1) = a Polynomial of degree 1: f(x) = a (x 0 ) + b (x 1 ) = a + b x or f(x) = k(x+A) Polynomial of degree 2: f(x) = a + b x + c x 2 or f(x) = k(x+A)(x+B)

Calc Ch.3.1 Conceptest 1

Calc Ch.3.1 Conceptest 1 answer

Calc Ch.3.1 Conceptest 2

Calc Ch.3.1 Conceptest 2 answer

Calc Ch.3.1 Conceptest 3

Calc Ch.3.1 Conceptest 3 answer

Calc Ch.3.1 Conceptest 4

Calc Ch.3.1 Conceptest 4 answer

Derivatives of exponential functions Think about the slope of f=e x How to calculate d(e x )/dx? Clues: d/dx(2 x ) ~ (0.6931) 2 x d/dx(3 x ) ~ (1.0986) 3 x ? d/dx(a x ) = (1.0) a x ? Guess:2 < a < 3 Ch.3.6:a=e=2.718…

Derivatives of exponential functions d/dx(e x ) = (1) e x Generalize the rule by recalling that ln(e) = 1 d/dx(e x ) = ln(e) e x Note that ln(2) = and ln(3) = d/dx(2 x ) = (0.6931) 2 x d/dx(3 x ) = (1.0986) 3 x d/dx(2 x ) = (ln2) 2 x d/dx(3 x ) = (ln3) 3 x d/dx(a x ) = ln(a) a x

Practice – Ch.3.2 Spend 5-10 minutes doing odd # problems on p.116