91Ó°ÊÓ

Find two numbers whose difference is 50 and whose product is a minimum.

Use Newton's method to find the point of intersection of the graphs to four decimal places of accuracy by solving the equation \(f(x)-g(x)=0 .\) Use the initial estimate \(x_{0}\) for the \(x\) -coordinate. f(x)=\sin x, g(x)=\frac{1}{5} x, \quad x_{0}=2

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ f(x)=x^{3}-6 x $$

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ h(t)=\frac{1}{3} t^{2}+\frac{3}{5} t^{5 / 3} $$

Approximate the zero of the function in the indicated interval to six decimal places. \(f(x)=\cos x-x\) in \(\left[0, \frac{\pi}{2}\right]\)

Use the Mean Value Theorem to prove that \(|\sin a-\sin b| \leq|a-b|\) for all real numbers \(a\) and \(b\).

Suppose that \(f\) and \(g\) are continuous on an interval \([a, b]\) and differentiable on the interval \((a, b)\). Furthermore, suppose that \(f(a)=g(a)\) and \(f^{\prime}(x)

Designing a Grain Silo A grain silo has the shape of a right circular cylinder surmounted by a hemisphere. If the silo is to have a volume of \(504 \pi \mathrm{ft}^{3}\), determine the radius and height of the silo that requires the least amount of material to build.

Traffic Flow Analysis The speed of traffic flow in miles per hour on a stretch of Route 123 between 6 A.M. and 10 A.M. on a typical workday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+52 \quad 0 \leq t \leq 4 $$ where \(t\) is measured in hours and \(t=0\) corresponds to 6 A.M. Sketch the graph of \(f\) and interpret your results.

Einstein's Theory of Special Relativity The mass of a particle moving at a velocity \(v\) is related to its rest mass \(m_{0}\) by the equation $$ m=f(v)=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} $$ where \(c\) is the speed of light. Sketch the graph of the function \(f\), and interpret your results.