91Ó°ÊÓ

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) \(f(x)=x^{2}+2 x, \quad x_{0}=1, \quad d x=0.1\)

Object dropped from a tower An object is dropped from the top of a \(100-\mathrm{m}\) -high tower. Its height above ground after \(t\) sec is \(100-4.9 t^{2} \mathrm{m} .\) How fast is it falling 2 sec after it is dropped?

Find the slope of the curve at the given points. \(y^{2}+x^{2}=y^{4}-2 x\) at \((-2,1)\) and \((-2,-1)\)

Find the derivatives of the functions in Exercises \(19-40\) $$ y=(4 x+3)^{4}(x+1)^{-3} $$

Find \(d p / d q\) $$ p=\frac{\sin q+\cos q}{\cos q} $$

A draining hemispherical reservoir Water is flowing at the rate of 6 \(\mathrm{m}^{3} / \mathrm{min}\) from a reservoir shaped like a hemispherical bowl of radius 13 \(\mathrm{m}\) , shown here in profile. Answer the following questions, given that the volume of water in a hemispherical bowl of radius \(R\) is \(V=(\pi / 3) y^{2}(3 R-y)\) when the water is \(y\) meters deep. a. At what rate is the water level changing when the water is 8 \(\mathrm{m}\) deep? b. What is the radius \(r\) of the water's surface when the water is y \(\mathrm{m}\) deep? c. At what rate is the radius \(r\) changing when the water is 8 \(\mathrm{m}\) deep?

Find \(d p / d q\) $$ p=\frac{\tan q}{1+\tan q} $$

Find the slope of the curve at the given points. \(\left(x^{2}+y^{2}\right)^{2}=(x-y)^{2} \quad\) at \(\quad(1,0)\) and \((1,-1)\)

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) \(f(x)=2 x^{2}+4 x-3, \quad x_{0}=-1, \quad d x=0.1\)

A growing raindrop Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop's radius increases at a constant rate.