Calculus Basics
Integration Applications
Modeling and Optimization
Parametric and Polar
Infinite Series
Differential Equations
100
What is the derivative of y=ln|sin2x| ?
2cotx
100
Find the area between y=sec²x and y=sinx from x=0 to x=pi/4
sqrt(2) /2
100
Find the linearization of f(x)=sqrt(2-x) at x=1
-x/2 + 3/2
100
Graph r=3cos(theta)
100
Find the Taylor series generated by f(x)=ex at x=0
1+x+x²/2+x³/6+.....+xn/n!
= sum_(n=0)^infinity xn/n!
100
Evaluate dy/dx = 4yx³ given the initial condition (2,1)
y=ex⁴-16
200
Evaluate the integral of (x²lnx) with respect to x
x3/3(lnx-1/3)+C
200
Find the volume of the solid generated by revolving the region bounded by y=sqrt(x), x=4, and the x-axis about the x-axis.
8pi
200
Find two numbers whose sum is 30 and whose product is as large as possible.
15, 15
200
Find the Cartesian equation of the line given by the parametric equations x=t/3 and y=2t-t²
y=6x-9x²
200
Express the repeating decimal as a geometric series and find the sum: 0.343434......
sum from n=1 to infiinity of 0.34(1/100)n-1
sum: 34/99
200
Use Eulerś Method with delta x=0.1 to find y when x=1.3 for dy/dx = y+x given y=2 when x=1.
y=3.024 when x=1.3
300
What is the derivative of y=xx ?
xxlnx+xx
300
Find the surface area of the solid formed by the region bounded by x=y², x=4, and the x-axis rotated around the x-axis.
pi/6 (17 sqrt(17) -1)
300
If r(x)=9x, c(x)=x³-6x²+15x, and x is units sold, what value of x will maximize profit?
About 3.414 units
300
Find the second derivative of the line given by the parametric equations x=ln|5t²| and y=2t³/3
(3/2)t³
300
Given 1/(1-x) = 1 + x + x² + x³ + ...... + xn + ..... use integration to find a power series to represent ln|1-x|
-sum, from n=0 to infinity, of (xn+1)/(n+1)
300
Evaluate dy/dx = -x/yex² given initial condition (0,1)
y=sqrt (e-x²)
400
Evaluate the integral of sin²xcos³x with respect to x
sin³x/3 -sin⁵x/5 +C
400
Evaluate the integral, from -infinity to infinity, of 1/(1+x²) w.r.t. x.
pi
400
A police cruiser is approaching a right-angled intersection from the south and is chasing a car that has turned the corner and is moving east. When the cruiser is 0.5 mi south of the intersection and the car is 1.2 mi east of it, a radar determines the distance between them is increasing at 20mph. If the cruiser is moving at 65mph at the instant of measurement, what is the speed of the car?
About 49mph
400
Find the area of the region enclosed by r=3(1+cos(theta))
27pi/2 units squared
400
Does the sum, from n=0 to infinity, of (2n+5)/3n converge?
Yes
400
A cake is cooling so it can be frosted. The cake must be cooled below 80oF before it can be frosted. The temperature in the bakery is at a constant 60oF. When the cakeś temperature is checked, it is at 100oF. 10 minutes later, it has dropped to 90oF. How much longer will it take for the cake to reach below 80oF?
About 14.094 min longer
500
Evaluate the integral of (2x+4)/(x³-2x²) w.r.t. x
2/x + 2ln |(x-2)/x| + C
500
Evaluate the integral of 1/(x²(sqrt(9-x²))) w.r.t. x
-1/9((sqrt(9-x²))/x) + C
500
Water runs into a conical tank at a rate of 6ft³/min. The tank stands point down and has a height of 10ft and a radius of 5ft. How fast is the water level rising when the water is 3ft deep?
About 2.094 ft/min
500
Find the length of half the asteroid x=cos³t, y=sin³t for 0<=t<=2pi
3
500
For what values of x does the power series
the sum, from n=1 to infinity, of (-1)n-1(x2n-1/2n-1)
=x-x³/3 +x⁵/5-...... converge?
-1<= x <= 1
500
Given dP/dt =0.0005 P(600-P) and P=10 when t=0, find P.
P=600/(59e-0.3t+1)