Unit 1
lim x(2x-5)(8x-1) / 8x3-1
x-> -infinity
2
f(x)= 2xsin2x, then f'(x)=
2sin2x+4xsinxcosx
Lim e2x / x2 is
x->∞
∞
∫2sin4x⋅cosxdx=
2/5 sin5x+C
Let R be the region in the first and fourth quadrant bounded by the graphs of x=y2 and x=9. What is the area of R?
36
if f(x)= {e2+ex for 0<x<2
{x2ex-xex for 2≤x<8
then lim f(x) is
x->2
2e2
What is lim 2(3+h)5-2(3)5 / h
h->0
810
If y=4x+6, what is the minimum value of the product of xy?
-3/4
Newton's Law of Cooling says that the rate of change of the temperature of an object T at the time t is directly proportional to the difference between the temperature of the object and the surrounding temperature Ts. What differentiable equation describes this relationship?
dT/dt= -k(T-Ts)
The base of a solid is the region bounded by the graph of y=x, y=-x, and x=5. For the solid, each cross-section perpendicular to the x-axis is a semicircle. What is the volume of the solid?
125π/6
if k≠0, then lim x2+3x-kx-3k / x2-k2 is
x->k
1
if f(x)=e5ln(x^2 -x), then f'(x)=
5(x2-x)4(2x-1)
Let f be a differentiable function such that f(4)=8 and f'(4)=-1. If the tangent line to the graph of f at x=4 is used to find an approximation to a zero of f, that approximation is
12
∫x4-x3-7x2+6x-9 / x-3 dx=
1/4x4+2/3x3-1/2x2+3x+C
What is the volume of the solid generated when the region bounded by y=x+sinx and the x-axis on the closed interval [0,2π] is revolved around the x-axis?
230.149
The graph f(x)= a-bx / c-x has a horizontal asymptote of y=-6 and a vertical asymptote x=6. What is the value of b+c?
12
If f(x)= tan-1(e2x), then f'(x)=
2e2x / 1+e4x
In a manufacturing plant, the cost of materials C(x)= 100/x decreases as the number of unites,x, increases. Also, the cost of labor L(x)=0.5x increases as the number of units increases. The total cost is the sum of the cost of materials and the cost of labor. Approximately how many units should be produced in order to minimize the total cost?
14
What is the general solution to the differential equation dy/dx=e2y/3x2 ?
-1/2e2y + 1/3x = C
What is the volume of the solid generated when the region in the first quadrant bounded by the graph y=√x, the x-axis, and the vertical line x=4 is revolved about the line y=3?
75.398
Lim x / sinx
x->0
1
Find the values of a and b such that h(x)= {ax2+7 x<1
{2x+b x≥1
differentiable at x=1. What is the sum of a and b?
7
A particle moves along the x-axis so that at time t>0 its position is given by x(t)=-1/3t3+3t2-8t+5. Over the interval 0<t<6, for what values of t is the speed of the particle decreasing?
0<t<2 and 3>t>4
The position of a ball rolling along a straight path is given by s(t), where t is time, in seconds, after the ball begins to roll. During the first 30 seconds of motion, the ball's velocity is proportional to the square root of the time from when the ball started to roll. What differentiable equation describes this relationship, where k is a positive constant?
ds/dt= k√t
What is the average value of y for the part of the curve y=-1/2x2+8 in the first quadrant?
16/3