Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline (Review and Velocity)Wednesday, January 11Recall that we covered interval notation and piecewise functions in class on Wednesday.Theory: Function CompositionA little trickier way of putting two functions together is function composition:(f ◦ g)(x) = f(g(x)).To evaluate f ◦ g at x, first evaluate g at x, and then plug this value into f.Examples: Function Composition1. Let f(x) = x2+ 1, g(x) = sin x, and h(x) = ex. What is f ◦ h? What is h ◦ (f ◦ g)?2. Write cos(ex) as a composition of functions.3. Write 2tan(x+2)as a composition of functions.Exponentials and LogarithmsThere are three Laws of Logarithms that you should always keep in mind when using thenatural logarithm function:ln(ab) = ln a + ln b ln(ab) = ln a − ln b ln(ab) = b ln aIn addition, eln(f(x))= f(x) and ln ef(x)= f(x).Also know that e0= 1, e1= e, ln e = 1, and ln 1 = 0.1Examples: Exponentials and Logarithms1. Simplifyln3(x + 1)2√1 − x.2. Simplifye3+x= 1.Theory: Trig functionsYou should commit the following right triangles to memory: (π/4, π/4, π/2)-triangle and the(π/3, π/6, π/2)-triangle.These triangles, with the help of the unit circle, will allow you to determine the sin θ andcos θ for multiples of θ = 0, π/6, π/4, π/3, π/2, andπ.Examples: Trig Functions1. Find the value of sin(2π/3).2. Find the value of cos(5π/4).Motivation: VelocityAverage velocities have a graphical interpretation in terms of slopes.In general, if we have a function y = f(x) and two points (x1, y1) and (x2, y2), the slopeof the line between these two points isy2− y1x2− x1and this slope is the average rate of change of y from x = x1to x = x2. This line is calledthe secant line of f(x) from x = x1to x = x2.This tells us how to compute average velocities. We’re really interested in instanta-neous velo city, not just average velocity. This is the velocity at an instant in time - inother words, the velocity that you see on the speedometer on your car. How do we calculatethat just from the distance traveled function?Let’s think about how you might approximate the instantaneous velocity at t = t1. Wecan calculate the average velocity from t = t1to t = x, and if x is close to t1, then average2velocity should be pretty close to the instantaneous velocity. In other words, the slope ofthis line, given bym =d(t1) − d(x)t1− x,should be very close to the instantaneous velocity when x is very close to t1.One might say “the instantaneous velocity is the limit of the functionm(x) =d(t1) − d(x)t1− xwhen x approaches
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