Math 1121 Calculus (II)

(1)

1. Find the area between the curves

(a) r= 2 cosθ, r = cosθ and the rays θ = 0, θ= 1 4 (b) r= 1

2sec² 1

2θ and the vertical line through the origin.

(c) r=e^θ, 0≤θ ≤π; r=e^θ, 2π≤θ ≤3π and the raysθ = 0, θ =π.

(d) r= tanθ, π

6 ≤θ ≤ π 3

2. Find the area of the following regions

(a) the region enclosed by one loop of the curver² = sin 2θ.

(b) the region lies inside the curver= 2 + sinθ and outside the curve r = 3 sinθ.

(c) the region lies inside both curvesr=asinθ, r=bcosθ, a >0, b >0.

(d) the region between a large loop and the enclosed small loop of the curver= 1 + 2 cos 3θ 3. Find the length of the curves

(a) r=θ², 0≤θ≤2π (b) r= 1/θ, π≤θ ≤2π

4. Find the Taylor polynomial of degree n for the given function f at the given pointa.

(a) f(x) = cosx, a= 0 (b) f(x) = lnx, a= 1, x >0

(c) f(x) = e^x², a= 0 (d) f(x) = √

x,a= 16 (e) f(x) = 1/x, a=−3

5. Letf(x) be an-degree polynomial in (x−a), sayf(x) =a0+a1(x−a) +· · ·+an(x−a)ⁿ. (a) Show thatP_k,a(x) = a₀+a₁(x−a) +· · ·+a_k(x−a)^k for any k ≤n.

(b) Show thatf(x) =P_k,a(x) for any k≥n.

(c) Show thatf(x) =Pk,b(x) for anyk ≥n and b∈R.

(2)

6. Let P(x) = c₀+c₁(x−a)ⁿ.

(a) By discussing the value of n (odd or even) and the sign of c₁ , determine whether P(x) has an extreme value at a.

(b) Let Q(x) = P(x) +c₂(x−a)ⁿ⁺¹ +c₃(x−a)ⁿ⁺². Determine whether Q(x) has an extreme value ata.

(c) Compare the result of problem (a) with the result of problem (b).

(d) Leth(x) = P(x) + (x−a)ⁿ⁺¹g(x) whereg(x) is continuous ata. Determine whether h(x) has an extreme value at a.

(e) Letk(x) = P(x)+q(x) whereq(x) is a function with lim

x→a

q(x)−q(a)

(x−a)ⁿ = 0. Determine whether k(x) has an extreme value at a.

7. Let f and g be two functions such thatf⁰(a), f⁰⁰(a), g⁰(a), g⁰⁰(a) and g⁰⁰⁰(a) exist.

(a) Prove that

x→alim

f(x)g(x)−P_2,a,f(x)P_3,a,g(x)

(x−a)² = 0.

(b) Determine whether

x→alim

f(x)g(x)−P_2,a,f(x)P_3,a,g(x)

(x−a)³ = 0.

is still true.