4-4 Delicious derivatives of Exponential & logarithmic functions

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Presentation transcript:

4-4 Delicious derivatives of Exponential & logarithmic functions

Derivative of e x Ex 1) Find

Derivative of a x Mini Proof

Ex 2) At what point on the graph of the function y = 2t – 3 does the tangent line have a slope of 21? deriv (4.921, 27.297)

Derivative of ln x Mini Proof

Ex 3) A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the value of m? (0, 0) and (a, ln a) m = deriv @ x = a same slope:

Derivative of log a x change of base Mini Proof

Power Rule (Reminder): *Now we will have any real # as our exponent* Ex 5a) Find b) Find

Ex 6) If f (x) = ln (x – 3), find f '(x). State the domain of f '(x). Domain? Thinking x  3 BUT domain of f ?  Can’t have ln of a neg # so domain of f : x – 3 > 0  (3, ) So domain of f '(x) is (3, ) also

Ex 8) Spread of flu modeled by equation a) Estimate the initial number of students infected with the flu. b) How fast is the flu spreading after 3 days? Need slope  derivative evaluated at t = 3

Ex 8) Spread of flu modeled by equation c) When will the flu spread at its maximum rate? What is this rate? Graph the derivative & find max 2nd Trace (CALC)  4: Max Max @ x = 2.999  3 days Rate? From part b  25 students/day (Zoom  0: zoomfit)

homework Pg. 175 #1, 10, 16, 20, 35, 39 Pg. 181 #1 – 41 (skip mult of 3), 49, 51