Unit 9:
                 Limits and Derivatives
                Involving Trig. Functions
                          (Chapter 2.5)
                           Dr. Gary Au
                       au@math.usask.ca
                      MATH123: Calculus I for Engineers
                        University of Saskatchewan
                                                     Unit 9: 1/13
  Motivation
        ◮ In the last unit, we have seen how to differentiate powers,
           as well as products, quotients, and compositions of those
           functions.
        ◮ Howabouttrig. functions? For instance, what’s the
           derivative of f(x) = sin(x)? Well...
                          f′(x)  = lim f(x +h)−f(x)
                                     h→0         h
                                 = lim sin(x +h)−sin(x).
                                     h→0           h
           This doesn’t look easy.
        ◮ In this unit, we learn some tools for computing trig. limits,
           then study the derivatives of trig. functions.
                                                                      Unit 9: 2/13
  Detour: Some Trig. Limits
     Question sin(x)
     Whatis lim      ? Does it even exist?
             x→0  x
     Fact
                            lim sin(x) = 1.
                           x→0   x
     Proof Idea        sin(x)
     Showthat cos(x) ≤   x   ≤sec(x) around x = 0, apply
     SqueezeTheorem.
                https://www.desmos.com/calculator/ghhcfjgchl
                                                         Unit 9: 3/13
  Detour: Some Trig. Limits
     Example1
     Evaluate lim cos(x)−1.
              x→0    x
                                                         Unit 9: 4/13