Multiple Choice
Problem 26
Problem 27
Problem 28
Problem 30
100
The average value of (secx)^2 on the interval [(pi/6),(pi/4)].
What is (12-4(sqrt3))/pi
100
GIFIREN
What is FIRE ENGINE
100
Write an integral expression for the area of S.
.5ʃ(1-2cosΘ)^2 dΘ, a=0 and b=pi/3
100
Find the interval of convergence of the power series for f.
(-2,0)
100
Write an equation for the line tangent to the graph g at x=1.
g'(x)=f'(x)[e^f(x)] g(1)=[e^f(x)]=e^2 g'(x)=-4(e^2) y-(e^2)=-4(e^2)(x-1) y=-4x(e^2)+4(e^2)+(e^2) y=-4x(e^2)+5(e^2)
200
Find the length of the arc of the curve defined by x=.5(t^2) and y=(1/9)(6t+9)^(3/2), from t=0 and t=2.
What is 8
200
Is f differentiable at x=0? Use the definition of the derivative with one-sided limits to justify your answer.
f is not differentiable because lim f(x) as x approaches 0 the negative side is 2/3, but the lim f(x) as x approaches 0 from the positive side is a negative value. Since f'(x) has a jump f is not differentiable.
200
Solve for dx/dΘ.
x=cosΘ -2[cos(Θ)]^2 (dx/dΘ) = -sin(Θ) +4cos(Θ)sin(Θ)
200
Find the sum of the series for the Taylor series f about x=-1.
1/[1-(x+1)] 1/(-x) 1/[-(-1)] = 1
200
HOROBOD
What is ROBIN HOOD
300
The function f is given by f(x)=(x^4)+4(x^3). On which of the following intervals is f decreasing.
What is (-3,0)
300
For how many values of a, -4_
Two. [f(b) - f(a)]/(b-a) [f(6) - f(a)]/(6-a) = 0 [1-f(a)]/(6-1) = 0 f(a)=1 a cannot equal 6 because it would does not exist.
300
Solve for dy/dΘ.
r=cos(Θ)-2cos(2Θ) rsin(Θ)=sin(Θ)-2cos(Θ)sin(Θ) (dy/dΘ)=cos(Θ)-2[cos(Θ)]^2+2[sin(Θ)]^2
300
TEEF FEET TEEF
What is TWO LEFT FEET
300
For -1.2
g'(x)=f'(x)[e^(f(x))] = 0 g'(x) changes signs when f'(x) does x=-1 and x=3 g'(x) changes from positive to negative at x=-1
400
AID AID AID ^
What is first aid
400
Is there a value of a, -4_
f'(c)=[f(b)-f(a)]/(b-a) f'(c)= [1-f(a)]/(6-a) f'(c)=1/3 a=3 If you look at the MVT formula after plugging in the knowns, for the bottom to equal 3 a must be 3.
400
Write an equation in terms of x and y for the line tangent to the graph of the polar curve at the point where Θ=pi/2. show the computation the lead to your answer.
(dy/dx)=[cos(pi/2)-2((cos(pi/2))^2)-2((sin(pi/2))^2)]/[-sin(pi/2)+4cos(pi/2)sin(pi/2)] =(0-0+2)/(-1+0)= -2 x=(1-2cos(pi/2))cos(pi/2) = 0 y=(1-2cos(pi/2))sin(pi/2) = 1 y-1=-2(x-0) y=-2x+1
400
Let g be the function defined by g(x)= ∫f(t)dt, a=-1 and b=x. Find the value of g(-1/2), if it exist, or explain why it doesn't.
ln2
400
The second derivative of g is g''(x)=(e^f(x))[(f'(x))^2+f''(x)]. Is g''(-1) positive, negative, or zero?
e^f(x) is always positive (f'(x))^2 is zero f''(x) is negative g"(-1) = pos[0+neg] g"(-1) is negative
500
Given the differential equation dz/dt=z(4-(z/100)), where z(0)=50, what is the limit of z(t) as t approaches infinity?
What is 400
500
The function g is defined by g(x)= ∫f(t)dt a=0 and b=x. On what intervals contained in [-4,6] is the graph of g concave up?
Concave up for [-4,0] and [3,6] because the second derivative is positive. g'(x)=f(x) g''(x)=f'(x) g''(x)>0 for [-4,0] and [3,6]
500
DID DID DID DID DID PUN DID DID DID DID DID
What is NO PUN INTENDED
500
Let h be the function defined by h(x) = f(x^2 -1). Find the first three nonzero terms and the general term of the Taylor series for h about x=0, and find the value of h(1/2).
h(1/2) = 1/[1-(1/4)] = 1/(3/4) = 4/3
500
Find the average rate of g', the derivative of g, over the interval [1,3].
[g'(b)-g'(a)]/(b-a) g'(3)= f'(3)e^f(3) = 0 g'(1)=f'(1)e^f(1) = -4(e^2) [0-(-4(e^2))]/(3-1) 4(e^2)/2 2(e^2)
M
e
n
u