The Ancient Origins of Derivatives
Derivatives are so Derivative
The Times They are A'Changin
Rates on Rates on Rates
100
What is the definition of the derivative of f, and when is a function differentiable?
A100
100
What is the derivative of y=x^2*sin(pi*x)
D100
100
The mass of the part of a metal rod that lies between it left end and a point x meters to the right is 3x^2 kg. What is the linear density when x is 1 meter? *Draw picture
T100
100
The altitude of a triangle is increasing at a rate of 1cm/min while the area of the triangle is increasing at a rate of 2 cm^2/min. At what rate is the base of the triangle changing when the altitude is 10cm and the area is 100cm^2?
R100
200
What is the derivative of the following function (using the definition of derivative)? f(x)=2x^2-16x+35
A200
200
What is the derivative of y=e^(x*sec(x))?
D200
200
Consider a blood vessel with radius 0.01cm, length 3cm, pressure difference 3000 dynes/cm^2, and viscosity n=0.027. What is the velocity gradient when r=0? v=P/(4nl)*(R^2-r^2) *Draw picture
T200
200
A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minutes after the woman starts walking?
R200
300
What is a function which is continuous, but not differentiable, at x=0? (And explain using limit laws)
A300
300
What are the equations of the tangent lines to the curve y=x*(5-x)^(1/2) (or y=x*sqrt(5-x) ) at points (1,2) and (4,4)?
D300
300
Boyle's Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV=C. Prove that the isothermal compressibility (how volume is changing with respect to pressure change) is given by B=1/P.
T300
300
Water is leaking out of an inverted conical tank at a rate of 10,000 cm^3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6m and diameter at the top of 4m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2m, what is the rate at which water is being pumped into the tank?
R300
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