Limits
What is the limit of cos(x/2) as x approaches π?
1
Find f'(x) for f(x) = (2x2+5)7.
28x(2x2+5)6
What is the integral of sec2(x)?
tan(x)+C
If P(t) is the size of a population at time t, which of the following differential equations describes linear growth in the size population?
a) dP/dt=200P
b) dP/dt=200
c) dP/dt=100t2
b) dP/dt=200
Name the vertical asymptote(s) of f(x).
f(x) = [x2+x-6] / [x2+7x+12]
x=-4
What is the limit of (x2-25)/(x-5) as x approaches 5?
10
If f(x)=(x3+4x2-12x+8)(3x2-9x+7), then find f'(1).
-4
What is the integral of x3√(4+x4)dx?
1/6(4+x4)3/2+C
Solve the equation:
dy/dx=5y2cos(x)
y=1/[-5sin(x)+C]
For -1.5<x<1.5, let f be a function with first derivative given by f'(x)=e[x^(4)-2x^(2)+1]-2. What are the intervals on which the graph of f is concave down? Use a calculator.
(-1.5,1) and (0,1)
What is the limit of 2+1/n as n approaches infinity?
2
Find f'(x) given f(x)=sin3(4x).
12sin2(4x)cos(4x)
A particle moves along the x-axis. The velocity of the particle at time t is 8t-t2. What is the total distance traveled by the particle from time t=0 to t=3?
27
Solve the differential equation dy/dx=2sin(x) with the initial condition y(π/3)=-1.
y=-2cos(x)
Determine the critical points of f(x)=8x3+81x2-42x-8.
x=-7, x=1/4
What is the limit of sin(x)/x as x approaches 0?
1
Find the equation of the tangent line to the curve f(x)=x2-10 passing through the point (5,1).
y-1=10(x-5)
Let R be a region in the first quadrant bounded above by the graph of y=(√x)+5 and below by the graph of y=x3. R is a base of a solid whose cross sections perpendicular to the x-axis are squares. What is the volume of the solid? Use a calculator
38.9668
Find the particular solution:
f''(x)=2x, f'(-3)=0, f(-3)=10
x3/3 -9x-8
A tank contains 30 liters of oil at t=3 hrs. Using a right Riemann sum with 3 subintervals and the data from the table below, what is the approximation of the number of liters of oil that are in the tank at t=15 hrs.
t(hours) | R(t) (liters/hr)
3 | 6.5
8 | 6.2
10 | 5.9
15 | 5.6
100.8
What is the limit of [ln(x+2) - ln(2)] all over x, as x approaches 0?
1/2
The equation gives the position s=f(t) of a particle moving on a coordinate line (s in meters, t in seconds).
s=6sin(t)-cos(t)
Find the particle's velocity as the time π/6 seconds.
3(√3)+1/2
Water is being pumped into a tank at a rate of r(t)=10(1-e-0.14t) gallons per minute, where t is the number of minutes since the pump was turned on. If the tank contained 200 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 35 minutes? Use a calculator
479 gallons
Find the particular solution to dy/dx=2xy2 given that there is a point (1,4) on the solution curve.
y=1/(-2x+9/4)
If f'(x)=√(x2+1) +x3+3, then f has local min at x=?
Use a calculator.
-1.7076