91Ó°ÊÓ

Use the rules for differentiating sums and differences, as in Example \(1,\) to compute the derivative of the given expression with respect to \(x\) $$ \sqrt{2}(\sin (x)-2 \cos (x))+2 x^{2}-3 / x $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=\sin (x), c=0, x=0.02 $$

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ \left.\frac{d f}{d x}\right|_{x=0}, f(x)=10 x-3 \sin (x) $$

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=6 t+3 $$

In Exercises 19-26, use the Reciprocal Rule to compute the derivative of the given expression with respect to \(x\) $$ 1 /\left(1+x^{2}\right) $$

Use the Quotient Rule to compute the derivative of the given expression with respect to \(x .\) $$ (1-\cos (x)) /(1+\cos (x)) $$

Calculate the linearization \(L(x)=f(c)+\) \(f^{\prime}(c), \cdot(x-c)\) for the given function \(f\) at the given value \(c\) $$ f(x)=e(x-1) / x, c=1 $$

Find a function whose derivative is the given function. \(3 / x^{3}\)

A function \(f,\) a point \(c,\) an increment \(\Delta x,\) and a positive integer \(n\) are given. Use the method of increments to estimate \(f(c+\Delta x)\). Then let \(h=\Delta x\) / \(N\). Use the method of increments to obtain an estimate \(y_{1}\) of \(f(c+h) .\) Now, with \(c+h\) as the base point and \(y_{1}\) as the value of \(f(c+h),\) use the method of increments to obtain an estimate \(y_{2}\) of \(f(c+2 h)\). Continue this process until you obtain an estimate \(y_{N}\) of \(f(c+N \cdot h)=f(c+\Delta x) .\) We say that we have taken \(N\) steps to obtain the approximation. The number \(h\) is said to be the step size. Use a calculator or computer to evaluate \(f(c+\Delta x)\) directly. Compare the accuracy of the one step and \(N\) -step approximations. $$ f(x) 1 / \sqrt[3]{x}, c=-8, \Delta x=1, N=4 $$

The leg of an isosceles right triangle increases at the rate of 2 inches per minute. At the moment when the hypotenuse is 8 inches, how fast is the area changing?