91Ó°ÊÓ

Find the rate of change of the area of a square with respect to the length of a side.

Find \(d^{3} y / d x^{3}\). $$ y=a x^{4}+b x^{3}+c x^{2}+d x+e $$

From the resulting graphs, guess a formula for \(f^{\prime}\). $$ f(x)=\frac{1}{2} x^{2} \text { for }-1 \leq x \leq 1 $$

a. Find the rate of change of the area of an equilateral triangle with respect to the length of a side. b. Let \(A(x)\) denote the area of the triangle when the length of a side is \(x\). Find a value of \(x\) for which \(A^{\prime}(x)=A(x) .\)

Find the fourth derivative of the function. $$ f(x)=3 x^{8}+\frac{3}{4} x^{6}-4 x^{3 / 4}+2 x^{-1} $$

Find an equation of the line tangent to the graph of \(f\) at the given point. $$ f(x)=x^{2}-3 x-4 ;(2,-6) $$

Let \(f(x)=x^{4}+x^{2}+8 x-1\), which has a zero in the interval \([-2,0] .\) Determine what happens when the Newt?n-Raphsôn meethod is used with the initial valuè of \(c\) equal to \(-1\).

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 \(d y / d x\). \(y^{5}\)

From the resulting graphs, guess a formula for \(f^{\prime}\). $$ f(x)=\sin x \text { for } 0 \leq x \leq 2 \pi $$

Let \(f(x)=\sqrt{x}-\frac{1}{2} .\) In trying to use the Newton-Raphson method to find a zero of \(f\), determine what goes wrong when we choose a. \(c_{1}=1\) b. \(c_{1}=4\)