Winter wk 6 – Tues.8.Feb.05 Calculus Ch.3 review: Polynomial rule for derivatives Differentiating exponential functions Chain rule and product rule 3.5 Trigonometric functions 3. 6 Applications of chain rule Energy Systems, EJZ

Differentiating polynomials and ex Integrating polynomials: Slope of ex increases exponentially: d/dx(ex) = ex d/dx(ax) = ln(a) ax

Review chain rule Differentiate exponential function y=eax Differentiate y=ex2=ez z=x2

Review product rule and quotient rule If y(x) = f(x) g(x) then Practice: y = x2 e-3x =f.g where f=______, g=______ If y(x) = f(x) / g(x) then

Differentiating trig functions Sketch the slope of y=sin(x) Does this look familiar?

Differentiating cosine Sketch the slope of y=cos(x) Does this look familiar?

Conceptest 1

Conceptest 1 solution

Conceptest 2

Conceptest 2 solution

(Hint: where is df/dx=0?) Conceptest 3 (Hint: where is df/dx=0?)

Conceptest 3 solution

(Hint: calculate the slope) Conceptest 4 (Hint: calculate the slope)

Conceptest 4 solution

Differentiating tangent tan(x) = sin(x)/cos(x) 1. Use identity sin2x+cos2x=1 to derive tan2x+1=_____ 2. Use product or quotient rule to find d(tan(x))/dx

Practice differentiating trig functions Drill on Ch.3.5 odd # problems through 41 (p.131) (skip #19) #5: y = sin(3x) = sin z where z=__________

3.6: Applications of the chain rule Finding the derivative of an inverse function (133) What is df/dx if f=x½? Trick: Write f2=x Differentiate Solve for df/dx= (Compare to result from polynomial rule.)

Using chain rule to find d/dx(lnx) Recall that elnx=x. Differentiate Practice p.136 on odd problems through #15

Using chain rule to find d/dx(ax) Recall that ln(ax)=x ln a. Differentiate Practice p.136 odd problems thru #35 (skip arc_ probs)

Chain rule for related rates Practice p.137 #46, 48