MATH 019 Lecture 9: Section 3.3 - The Product and Quotient Rules

5. Suppose g is a function with continuous derivatives, and that go)-3,g'(o)-2,g"0)-1,and g" (0)--3. a) What is the Taylor polynomial of degree 2 for g, centered at a o? b) what is the Taylor polynomial of degree 3 for g, centered at a=0 ? c) Use石(x) and (x) to approximate g(0.1). 6. For the function f(x)-Inx: a) List the first four derivatives of f(x). b) What are the values of the first four derivatives of f(x) evaluated at a-1? c) Write T,(x) as a polynomial in the "long form: A+Blx-o)+Cx-o+Ex-o d) Optional Challenge question: Write T,(x) using summation notation. e) Graph f(x),五(x),万(x), (x), and (x) on the same set of axes, with-Q5cxs2 and-5cy

37, find the equations of the tangent and to the graph of the function at the given point. In Exercises 32 32. f(x) =x3-rat x = 1 33. f(t)-e' + 3 at t 0 34, g(x)=inxatx_ 1 35, f(x) = 4 sin x at x = π/2 36, f(x) =ー2 cos xatx = π /4 37, f(x) = 2x + 3 at x-5 Review 38. Given that e1, approximate the value of e0.1 using the tangent line to f(x) = ex at x = 0.

8. Use Part II of the Fundamental Theorem of Calculus to evaluate the following integrals or explain why the theorem does not apply: (a) (45 +2+2 +1)dz (b)da (c) sec (t) dt [/3sisin /e 2) da π/3 sin(z) T/2 (e) (sin(a)+ 5 cos(x) da 1+Va 11下 (g)V