Derivatives
Integrals
Related Rates
Taylor Mac
Particle Motion
100
What is the derivative of 3x^2?
6x
100
What is ∫(5x+5)dx?
(5/2)x^2+5x+C
100
What is Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm?
(1/80π)cm/min
100
Write the first three nonzero terms and the general term of the Maclaurin series for sin(x)
x-(x^3)/(3!)+(x^5)/(5!)
100
What is the acceleration of the vector V(t)= <5x^2,4x>?
<10x,4>
200
Derivative of sin(2x)
2cos(2x)
200
What is ∫(sin(2x))dx?
-(1/2)cos(2x)+C
200
A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?
What is .1319 ft/sec
200
Write the first four nonzero terms of the Taylor series for ( ) 2 sin x about x = 0.
(x^2)-(x^6)/(3!)+(x^10)/(5!)-(x^14)/(7!)
200
Find the speed of a particle at t=3. x'(t)=4t+1 y'(t)=sin(t^2)
13.006
300
What is the derivative of (2x)(4x^2)
2(4x^2)+(2x)(8x)
300
What is ∫(cos(2x)^2)dx
sin(2x)/4+C
300
Water is being poured into a conical reservoir at the rate of pi cubic feet per second. The reservoir has a radius of 6 feet across the top and a height of 12 feet. At what rate is the depth of the water increasing when the depth is 6 feet?
What is 1/9 ft/sec
300
Write the first three nonzero terms of the Maclaurin series ln(1+x^3)
What is x^3-(x^6/2)+(x^9/3)
300
A particle is moving along a curve so that x(t)= t^2-4t+8 and y'(t)= te^(t-3)-1. Find the time t, 0 4, ≤ ≤t when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time?
What is t= 2.2079 and the particle is moving right
400
If f(x)= ln(x+4+e^(-3x)) then f'(0)
What is 1/5
400
What is ∫(3x+1)^5dx
What is (3x+1)^6/18 + C
400
A camera is located 50 feet from a straight road along which a car is traveling at 100 feet per second. The camera turns so that is pointed at the car at all times. In radians per second, how fast is the camera turning as the car passes closest to the camera?
What is -2 rad/sec
400
The Maclaurin series for the function f is given by f(x)= ∑((-1)^n(2x)^n)/n-1 Find the interval of convergence for the Maclaurin series of f
What is (-1/2,1/2>
400
A particle is moving along a curve so that x(t)= t^2-4t+8 and y'(t)= te^(t-3)-1. There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following. (i) The two values of t when that occurs (ii) The slopes of the lines tangent to the particle’s path at that point
What is t=1 and t=3 and slope at t=1 is .432 and slope at t=3 is 1
500
Let P(x)=3x^2-5x^3+7x^4+3x^5 be the fifth degree Taylor polynomial for the fraction f about x=0. What is the value of f'''(0)
What is -30
500
What is ∫xsin(6x)dx=
What is -(x/6)cos(6x)+(1/36)sin(6x)+C
500
A balloon leaves the ground 500 feet away from an observer and rises vertically at the rate of 140 feet per minute. At what rate is the angle of inclination of the observer's line of sight increasing at the instant when the balloon is exactly 500 feet above the ground?
What is 7/50 rad/sec
500
Let f be the function given by f(x)= sin(5x+(pi/4)), and let P(x) be the third degree Taylor polynomial for f about x+0. Find the coefficient of x^22 in the Taylor series for f about x+0
What is (-5^(22)√2)/(2(22!))
500
A particle is moving along a curve so that x(t)= t^2-4t+8 and y'(t)= te^(t-3)-1. There is a point with x-coordinate 5 through which the particle passes twice. Find the y coordinate of that point, given y(2)= 3+(1/e)
What is 4
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