Water flows into a conical tank at a rate of 2 ft³/min. If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.

Use the definition of the derivative to evaluate the following limits.

${\lim }_{x\to 2}{\frac{5^x-25}{x-2}}_{}$

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

b. Interpret the meaning of your results in part (a).

A general proof of the Chain Rule Let f and g be differentiable functions with h(x)=f(g(x)). For a given constant a, let u=g(a) and v=g(x), and define H (v) = <1x1 matrix>

c. Show that h′(a) = lim x→a ((H(g(x))+f′(g(a)))⋅g(x)−g(a)/x−a).

A spherical balloon is inflated at a rate of 10 cm³/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?

Let f(x) = x².

a. Show that f(x)−f(y) / x−y = f′(x+y²), for all x≠y.

Derivatives of even and odd functions Recall that f is even if f(−x) = f(x), for all x in the domain of f, and f is odd if f(−x) = −f(x) for all x in the domain of f.

b. If f is a differentiable, odd function on its domain, determine whether f' is even, odd, or neither.