91Ó°ÊÓ

Find the fourth derivative of the function. $$ f(x)=\sin x-\cos x $$

Find an equation of the line tangent to the graph of \(f\) at the given point. $$ f(x)=(x+1)(x-1) ;(3,2) $$

a. Find the rate of change of the volume of a sphere with respect to the radius. b. What relationship does the rate of change bear to the surface area of the sphere?

Suppose that \(y\) is a differentiable function of \(x\). Express the derivative of the given function with respect to \(x\) in terms of \(x, y\), and \(d y / d x\). \(y^{-2 / 3}\)

Let \(f(x)=x^{3}-x\), and suppose we attempt to approximate a zero of \(f\) in \((0,1)\). Determine what happens when the Newton-Raphson method is used with the initial value of \(c\) equal to \(1 / \sqrt{5}\).

Find an equation of the line tangent to the graph of \(f\) at the given point. $$ f(x)=\sin x-\cos x ;(\pi / 2,1) $$

Suppose that \(y\) is a differentiable function of \(x\). Express the derivative of the given function with respect to \(x\) in terms of \(x, y\), and \(d y / d x\). \(2 / y\)

Find the fourth derivative of the function. $$ f(x)=\sin \pi x $$

When radioactive iodine I \(^{128}\) disintegrates, the rate of change of the amount \(A(t)\) present at time \(t\) is proportional to the amount at that time. The constant of proportionality is \(-0.028\). a. Write an equation that relates \(A(t)\) and the rate of change in \(A(t)\). b. Suppose \(A(1)=3\). Find \(A^{\prime}(1)\).

Find the fourth derivative of the function. $$ f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e $$