Problem 3
In Exercises \(1-8,(a)\) state whether or not the function satisfies the hypotheses of the Mean Value Theorem on the given interval, and (b) if it does, find each value of \(c\) in the interval \((a, b)\) that satisfies the equation $$f(x)=x^{1 / 3} \quad \text { on }[-1,1]$$
Problem 3
\(f(x)=x+\frac{1}{x}, \quad a=1\)
Problem 3
Volume The radius \(r,\) height \(h,\) and volume \(V\) of a right circular cylinder are related by the equation \(V=\pi r^{2} h .\) (a) How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? (b) How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? (c) How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?
Problem 3
In Exercises \(1-6,\) use the First Derivative Test to determine the local extreme values of the function, and identify any absolute extrema. Support your answers graphically. $$y=2 x^{4}-4 x^{2}+1$$
Problem 4
In Exercises \(1-6,\) use the First Derivative Test to determine the local extreme values of the function, and identify any absolute extrema. Support your answers graphically. $$y=x e^{1 / x}$$
Problem 4
\(f(x)=\ln (x+1), \quad a=0\)
Problem 4
Finding Area Show that among all rectangles with an 8-m perimeter, the one with largest area is a square.
Problem 4
Electrical Power The power \(P\) (watts) of an electric circuit is related to the circuit's resistance \(R\) (ohms) and current \(I\) (amperes) by the equation \(P=R I^{2} .\) (a) How is dP/dt related to \(d R / d t\) and \(d I / d t\) ? (b) How is \(d R / d t\) related to \(d I / d t\) if \(P\) is constant?
Problem 4
In Exercises \(1-8,(a)\) state whether or not the function satisfies the hypotheses of the Mean Value Theorem on the given interval, and (b) if it does, find each value of \(c\) in the interval \((a, b)\) that satisfies the equation $$f(x)=|x-1| \quad \text { on }[0,4]$$
Problem 5
\(f(x)=\tan x, \quad a=\pi\)
