Derivatives
Integrals
Limits
Particle in Motion
Hodge Podge
100
Derive
f(x) = x3+x2+x+100
What is
f'(x) = 3x2+2x+1
100
The formula for the Disc Method is:
, where f(x) is the function that is further from the axis of rotation and g(x) is the function closer to the axis of rotation.
100
Find limx→ 4 (3x2+8x)/(x+4)
What is 10?
100
For t>0, a particle moves along the x-axis. The velocity of the particle at time t is given by
v(t)= 1+2sin((t^(2))/(2)).
The particle is at position x=2 at time t=4
At time t=4, is the particle speeding up or slowing down?
Slowing down, the acceleration and velocity are opposite signs.
100
How do you determine the concavity of a function?
Take the second derivative of the function, set the function equal to zero, and determine the points of inflection.
200
Derive
f(x) = ln(x)cos(2x)
What is
f'(x) = (-sin(2x)2)/(x)
200
If p(x) is the rate at which potato chips are being made in a factory, explain what the integral of p(x) over the interval [0,4] means in the context of the problem.
What is
The integral of p(x) over the interval [0,4] represents the total number of potato chips that are made after 4 hours.
200
What is the definition of a limit?
lim x→a f(x) = L, where L is the value that x become as it approaches a chosen value a.
200
For t>0, a particle moves along the x-axis. The velocity of the particle at time t is given by
v(t)= 1+2sin((t^(2))/(2)).
The particle is at position x=2 at time t=4
Find all times during 0<t<3 when the particle changes direction.
What is 2.707
200
If f(x)=x2, find a derivative of f-1(x) at x=9.
What is 1/6?
300
Derive
f(x) = (ln(x^3-4x))/(x^3-4x)
What is
−((3x^(2)−4)(ln(x^(3)−4x)−1))/(x^(2)(x−4)^2)
300
Evaluate the indefinite integral of f(x)= (4x+5)e2x^2+5x+3
What is e2x^2+5x+3 + C
300
Find limx→ 4 (x2-2x-8)/(x-4)
What is 6?
300
For t>0, a particle moves along the x-axis. The velocity of the particle at time t is given by
v(t)= 1+2sin((t^(2))/(2)).
The particle is at position x=2 at time t=4
Find the position of the particle at time t=0.
What is
-3.815
300
The derivative of 10=3x+2xy+2y
What is
(3+2y)/(-2x+8y)
400
Derive
(x^(3)+20x^(2)-5x)/(x^2+x)
What is
−48/((x+1)^(3))
400
Calculate the integral of the function f(x)= 12x3+ 6x2 +2x+1 over the interval [0,3] (NO CALCULATOR)
What is 309?
400
find lim h→ 0 of ((7x+7h)-7x)/h
What is 7?
400
For t>0, a particle moves along the x-axis. The velocity of the particle at time t is given by
v(t)= 1+2sin((t^(2))/(2)).
The particle is at position x=2 at time t=4
Find the total distance the particle travels during t=0 to t=3.
What is
5.301
400
A circular pool of water is expanding at a rate of 16π in.2/sec. At what rate is the radius expanding when the radius is 4 inches?
What is 2 in./sec?
500
Derive
(sin(e^(2x))ln(sqrt(2x)))/((x^(2)-3)(sqrt(sec(2x))))
What is
(x^(2)−3)(sin(e^(2x))+2xe^(2x)(ln(x)+ln(2))cos(e^(2x)))−x(x^(2)−3)(ln(x)+ln(2))tan(2x)sin(e^(2x))−2x^(2)(ln(x)+ln(2))sin(e^(2x))/((2x(x^(2)−3)^(2)√sec(2x))
500
Find the area between the two curves of f(x)=x2 and f(x)= x+2 (CALCULATOR ALLOWED)
What is 4.5?
500
Evaluate limx→ ∞ (5x2+4x+3)/(6x2+8x-14)
What is 5/6?
500
For 0≤t≤6, a particle is moving along the x-axis. The particle’s position, x(t), is not explicitly given. The velocity of the particle is given by v(t)=2sin(e^(1/4))+1. The acceleration of the particle is given by a(t)=(1/2)(e^(t/4))cos(e^(t/4)) and x(0)=2
Find the total distance traveled by the particle from time t=0 to t=6.
What is 12.573
500
Approximate the area under the curve y=x3 from x=2 to x=3 using four left-endpoint rectangles.
What is 13.953?