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Derivatives and Applications
Averages
Integrals
Differential Equations
Area and Volume
100
The coordinates of the point on the curve y=x2+1 which is closest to (3,1) is
What is (1,2)
100
The temperature outside a house during a 24-hour period is given by Where F(t) is measured in degrees Fahrenheit and t is measured in hours. Find the average temperature, to the nearest degree Fahrenheit, between t=6 and t=14
What is ≈ 87º F
100
If: [ x -- 2 - 5 - 7 - 8 ] [ f(x) -- 10 - 30 - 40 - 20 ] The function f is continuous on the closed interval [2,8] and has values that are given in the table above. Using the sub-intervals [2,5], [5,7], and [7,8], what is the trapezoidal approximation of the "integral of f(x) from 2 to 8 in terms of x"?
What is 160
100
At any time t ≥0, in days, the rate of growth of a bacteria population is given by y’=ky, where y is the number of bacteria present and k is a constant. The initial population is 1,500 and the population is quadrupled during the first 2 days. By what factor will the population have increased during the first 3 days?
What is 8
100
If the region bounded by the curve f(x)=sec x, the x-axis, y-axis, and the line x=π/4, is revolved about the x-axis, what is the volume of the resulting solid?
What is π
200
The function y=x4+bx2+8x+1 has a horizontal tangent and a point of inflection for the same value of x. What must be the value of b?
What is -6
200
a piece of steel at 1500 degrees F is removed from the oven and placed in a room at 70 degrees F. The temperature T of the steel, t minutes after it starts cooling, is given by T = 70 + 1430e^(-0.3t). Find, to the nearest degree, the average temperature of the steel over the first hour.
Avg = 149.444443235
200
get the integral of (144+144x+36x^2)^1/2 dx
What is 36 (x^2 + 4x + 4)
200
The function y satisfies a differential equation of the form y' = ky for some number k. If you are told that when t = 3 that y is 3 and the rate of change of y is 5 then what is k?
What is 5/3
200
The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x+2y=8. If the cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?
What is 16.755
300
The volume of an expanding sphere is increasing at a rate of 12 cubic feet per second. When the volume of the sphere is 36π cubic feet, how fast, in square feet per second, is the surface area increasing?
What is 8 feet2/sec
300
Let f(x) = (14 π)x^2 and g(x)= k^2sin(pi x/2k) for k>0 Find the average value of f on [1,4].
What is 3
300
∫ from 0 to 2 of √((x^2)-4x+4) (square root is over all)
What is -2
300
Consider the differential equation: dy/dx = (3-x) / y let y = f(x) be the particular solution to the given differential equation for 1 < x < 5 such that the line y = -2 is tangent to the graph of f. Find the x-coordinate of the point of tangency, and determine whether f has a local maximum, local minimum, or neither at this point.
What is Radius r = 2. If a tangent line is horizontal
300
Let R be the region in the first quadrant enclosed by the graphs of y = 2x and y = x^2 Find the area R
What is 4/3 square units
400
A rectangular plot of farmland that is bounded on one side by a river and is to be fenced in on the other three sides by an electric fence. With 1800 feet of wire at your disposal, what is the largest area you can enclose?
What is 405,000 square ft
400
The number of hours, H , of daylight in Madrid as a function of date is approximated by the formula H=12+2.4sin(0.0172(t−80)) where t is the number of days since the start of the year. (We can think of t=0 as the stroke of midnight on Dec. 31/Jan 1; thus, January falls between t=0 and t=31 , February falls between t=31 and t=59 , etc.). Find the average number of hours of daylight in Madrid (assuming in each case that it is not a leap year): A. in January: average hours = B. in June: average hours = C. over a year: average hours =
Integrate the function H = 12+2.4sin(0.0172(t−80)) over the interval and divide by the length of the interval (jan = 31 and jun = 30) primitive of H is G = 12.x - (2.4/0.0172).cos(0.0172(t-80)) + c (where c = constant)
400
integration of (3x+y)^-2 dy ? (You must integrate in terms of y)
What is 9(x^2)y + y^3/3 + (6xy^2)/2 + C
400
Consider the differential equation: dy/dx = (3-x) / y Let y = g(x) be the particular solution to the given differential equation for -2 < x < 8, with the initial condition g(6) = -4. Find y = g(x).
What is r
400
Find the area of the region bounded by the curves. y = cos x, y = sin2x, x = 0, x = pi/2
What is 1/2
500
A rectangular sheet of paper with a perimeter of 36cm and dimensions “x”cm and “y”cm is to be revolved around one of the sides y to sweep around into a cylinder. What values of x and y will give the greatest volume?
What is X=12cm Y=6cm
500
A train travels along a track at a constant speed of 50 miles per hour from 12:00 noon to 1:00 pm. At 1:00 pm it slows gradually and comes to a halt at 1:15 PM. It travels a distance of 6 miles between 1:00 PM and 1:15 PM. What was the average speed of the train between 12:00 noon and 1:15 PM? The average speed of the train in miles per hour was
av. speed = 56.25/1.25 = 45 mph
500
Water is pumped out of a holding tank at a rate of 5 - 5e^-0.15t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later? The answer has to be rounded to one decimal place. The amount of water in the tank after one hour = what??
What is 733.329
500
Consider the differential equation dy/dx = (5x^2) - [6/(y-2)] for y ≠ 2. Let f(x) be the particular solution to this differential equation with the initial condition f(-1) = 4. Is it possible for the x-axis to be tangent to the graph of f at some point?
What is Never
500
Let S be the solid described as follows: The base of S is the region in the xy plane bounded by the curves y=e^x,y=2 ,x=2 ,x=3 , The cross-sections of S with a plane through the point x and perpendicular to the x axis are square. Find the area A(x) of a cross-section of S with a plane through a point x and perpendicular to the x axis. Answer (formula) A(x)= ____________
What is (e^x - 2)²