2, 3, 4, 5 ,6 Calculus MTH 202-002 Fall 2015 Final M. Sayrafiezadeh Do any 5 out

Question

2, 3, 4, 5 ,6

Calculus MTH 202-002 Fall 2015 Final M. Sayrafiezadeh Do any 5 out of 6 problems Verify your answers as much as possible To receive credit show all your work clearly 1. Evaluate the limit, x2-6x (a) lim 2. Find the linear approximation L(x) of the given function around the given point a. (a) f(x) = x. _ 3x2 , a-| (b) Find the error in the approximation when x 1.01 4-12 () lim (c) f(x)-x cosx: + 1, a = 0 (d) Find the error in the approximation when x 0.01. 3. (a) Find all critical points of the function f(x)-3-16x' +18x (b) Determine the local maximum and the minimum Derivative Test.) (c) Find all the inflection points. points.(Use the First or the Second 4. An open top box is to have a bottom with one edge 3 times as long as the other. Its volume has to be 144 . To save cost, they want the surface area to be minimum. Since you know calculus, they have come to you for help. Find the dimensions of the box that will make the surface area minimum 5. Let 5 Let fx)+4 Use logarithmic differentiation to determine, (a) f(x) b) f(a) 6. We want to solve the equation cos.xx (a) Let f(x)-cos.x-x. Show that f(x) has a root between 0 and 2 (b) Use Newton's method with the initial value 0.5 to find the root of the equation correct to at least 8 decimal places. (c) Use Rolle's Theorem to show that the equation has exactly one solution.

Explanation / Answer

a) f(x)=x4 -3x2, a=1

f '(x)=4x3 -6x

f(a)=f(1)=14 -3*1=-2

f '(a)=f '(1)=4*13 -6*1=-2

linear approximation

f(x)= f(a)+ f '(a)(x-a)

f(x)=-2-2(x-1)

f(x)=-2-2x+2

f(x)=-2x

error:

f(x)- f(a)= f '(a)(x-a)

f(1.01)-f(1)= -2(1.01-1)

f(1.01)-f(1)= -0.02

b)

f(x)=xcosx2+1, a=0

f '(x)=(1cosx2-xsinx22x)+0

f '(x)=(cosx2-2x2sinx2)

f(a)=f(0)=0cos02+1=1

f '(a)=f '(0)=(cos02-2*02sin02)=(1-0)=1

linear approximation

f(x)= f(a)+ f '(a)(x-a)

f(x)=1+1(x-0)

f(x)=x+1

error:

f(x)- f(a)= f '(a)(x-a)

f(0.01)-f(0)= 1(0.01-0)

f(0.01)-f(0)=0.01

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