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 subintervals [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
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
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 ft2
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
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