Rates of Change
Areas
Graphing Calculus
Related Rates
Optimization
100
Find the rate of change for the following function:
a) from x=0 to x=2
b) at x=2
a) e^2 - 1
b)2e^2
100
A boat consumes fuel at a rate of
liters per hour. Find the amount of fuel consumed in the first two hours. Round to two decimal places.
9.14 liters
100
Determine the critical points of f(x)= 8x³ + 81x² - 42x + 8
-7, .25
100
A person is standing 350 feet away from a model rocket that is fired straight up into the air at a rate of 15 ft/sec. At what rate is the distance between the person and the rocket increasing 20 seconds after liftoff?
9.76187 feet/sec
100
We have 45 m^2 of material to build a box with a square base and no top. Determine the dimensions of the box that will maximize the enclosed volume.
l = w = 3.8730
h = 1.9365
200
Suppose that the volume of water in a tank is given by the function
does the water level in the tank ever stop changing? If so, at what t-value?
It does, at t=2.5 seconds.
200
Find the right Riemann sum over the interval [0,4] using 4 rectangles on the following function:
f(x) = 5/x
125/12
200
Find the x-values where the following function has an absolute max and min on the interval: [-4,2]
f(x)= ln(x² + 4x + 14)
min: x=-2
max: x=2
200
Two people are at an elevator. At the same time one person starts to walk away from the elevator at a rate of 2 ft/sec and the other person starts going up in the elevator at a rate of 7 ft/sec. What rate is the distance between the two people changing 15 seconds later? Round your answer to 4 decimal places.
7.2801 ft/sec
200
A farmer has 6800 ft. of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?
3400 feet by 1700 feet
300
Find the x value(s) where f(x) is parallel to the x-axis:
x = 1, -1
300
Find the left Riemann sum over the interval [2,8] using 3 subintervals for the following function:
40
300
Determine the intervals where the following function is increasing/decreasing and the x-values where it has a local max/min.
y = 2x³ - 10x² +12x -12
Increasing:
Decreasing:
Local Max:
Local Min:
300
A person is 500 feet away from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 15 ft/sec. At what rate is the angle of elevation, θ, changing when the hot air balloon is 200 feet above the ground. Round your answer to 5 decimal places.
-0.02586 degrees/second
300
We want to build a box whose base length is 6 times the base width and the box will enclose 20 in3. The cost of the material of the sides is $3/in2 and the cost of the top and bottom is $15/in2. Determine the dimensions of the box that will minimize the cost.
w = 0.7299 L = 4.3794 h = 6.2568
400
Write the equation for the tangent line parallel to the line 12x - 2y = -64 on the function:
y=6x+11
400
The water level of a strait is changing at a rate of
centimeters per hour (where t is the hours since midnight, round your answer to 1 decimal place).
2.8 cm/hr
400
Determine the intervals where the following function is CCU/CCD as well as the x-values of any inflection points.
f(x) =
CCU:
CCD:
IP:
400
At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 24 knots and ship B is sailing north at 20 knots. How fast (in knots) is the distance between the ships changing at 6 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
31.028009935065 knots
400
A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=1-x². What are the dimensions of such a rectangle with the greatest possible area?
w = 1.155
h = .667
500
A potato is launched vertically upward. The distance in feet that the potato travels from the ground after t seconds is given by
find the velocity of the potato upon hitting the ground.
-116 feet per second
500
Suppose that t months from now the population of a town will be growing at the rate of
people per month. The initial population is 5000. What will be the population in 3 years?
8240 people
500
Determine the increasing/decreasing intervals, x-values of local max/mins, CCU/CCD intervals, and x-values of inflection points for the following function:
Increasing: (-oo, 0) u (8/7,oo)
Decreasing: (0,8/7)
Max: x=0
Min: x= 8/7
CCU: (2/7,oo)
CCD: (-oo,2/7)
IP: x = 2/7
500
A tank of water in the shape of a cone is being filled with water at a rate of 12 m3/sec. The base radius of the tank is 26 meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing when the radius of the top of the water is 10 meters?
h′=3/(25pi) m/s
500
We have a piece of cardboard that is 50 cm by 20 cm and we are going to cut squares out of the corners and fold up the remaining sides in order to form a box. Determine the height of the box that will give a maximum volume.
h = 4.4018 cm