Slope Detectives
This is the derivative of f(x)=5x3
What is f'(x)=15x2
A farmer has 120 feet of fencing to enclose a rectangular pasture adjacent to a long stone wall. If she does not need to fence the side along the wall, what is the maximum area, in square feet, she can enclose?
What is 1800 square feet?
The general solution to the differential equation:
dy/dx = 4x3y
for y>0.
What is y = Ce(x^4)
This type of Riemann Sums uses the left endpoint of each subinterval
What is Left Riemann Sum
This is the derivative of f(x)=(7x4)-(2x2)
What is f'(x)=28x3-4x
A manufacturer makes an open-top box by cutting equal squares of side length x from the corners of a 10-inch by 16-inch sheet of cardboard and folding up the sides. This value of x, in inches, maximizes the volume of the box.
what is 2 inches?
The general solution to the differential equation:
dy/dx = 3x2/2y
with the initial condition y(0) = -2.
What is y = -√(x3+4)
This is the width of each subinterval when using 5 right Riemann sums of equal width to evaluate from (0,10)
What is 2
This is the derivative of f(x)=(3x2)(4x) using product rule
What is f'(x)=36x2
A rectangular poster must contain 50 square inches of printed text. The margins at the top and bottom are each 4 inches, and the margins on the left and right sides are each 2 inches. This is the total width, in inches, of the poster that minimizes the overall area.
what is 9 inches?
The general solution to the differential equation:
dy/dx = x2(1+y2)
y = tan(1/3x3+C)
For f(x)=x on [0,2] using 2 subintervals and left endpoints, this is the value of the Riemann sum.
What is 1?
This is the derivative of f(x)=(x2)/(x+1) using quotient rule
What is f'(x)=(x2+2x)/((x+1)2)
A right circular cylinder is inscribed inside a cone with a height of 12 inches and a base radius of 6 inches. This is the radius of the cylinder, in inches, that yields the maximum possible volume.
what is 4 inches
The particular solution to y = f(x) to the differential Equation:
dy/dx = cos(x)/ey
With the initial condition y(π) = 0
What is y = ln(sin(x)+1)
This is the value of a left Riemann sum approximation for the area under the function f(x) = 2x + 3 on the interval [1, 5] using 4 subintervals of equal width.
What is 32?
This is the derivative of f(x)=(x2+1)(x2−3x+4)
An oil refinery is located on the bank of a straight river that is 2 miles wide. A storage tank is located on the opposite bank, 6 miles downstream. A pipeline must be laid from the refinery to the tank. It costs $50,000 per mile to lay pipe underwater, but only $30,000 per mile to lay it along the land. This is the distance downstream, in miles, where the pipeline should hit the opposite bank to minimize total cost
what is 1.5 miles?
The particular solution y = f(x) to the differential equation:
dy/dx = 2x/y-3
with the initial condition y(2) = 1
What is y = 3 - √(2x2-4), for x>0
The value of a trapezoidal sum approximation for the area under the function f(x) = 2x + 3 on the interval [1, 5] using 4 subintervals of equal width?
What is 36