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)
(NO CALC)
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? (EXTRA 100 pts... say why)
(CALC ALLOWED)
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) = [(1/x)*cos(2x)] - [sin(2x)*2*ln(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 limit definition of a derivative?
(See overhead)
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.
(CALC ALLOWED)
What is 2.707
200
The formula for finding derivatives of inverse functions
What is (See overhead)?
300
Derive
f(x) = (ln(x^3-4x))/(x^3-4x)
What is
(See overhead)
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)
(NO CALC)
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.
(CALC ALLOWED)
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
(see overhead)
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
(NO CALC)
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.
(CALC ALLOWED)
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)))
What is
[cos(e^(2x)*e^2x*2ln(sqrt(2x))] +
[1/2*1/(2x)*2*sin(e^(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)
(NO CALC)
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.
(CALC ALLOWED)
What is 12.573
500
Approximate the area under the curve y=x3 from x=2 to x=3 using four left-endpoint rectangles.
(CALC ALLOWED)
What is 13.953?