Prepared: Sat Oct 30 12:26:00 CDT 1999

MATH 1271

QUIZ V

1) Write an equation of the tangent line to the curve defined by

at the point .

To write down the equation of a line a point on the line and the slope will be enough. We have the point, so we need the slope only.

The slope of the tangent line to the function at the point is the derivative of f evaluated at 1, i.e. f'(1).

and

So the equation of the tangent line is

2) A circular oil slick of uniform thickness is caused by a spill of 1 of oil. The thickness of the oil slick is decreasing at the rate of 0.1 cm/h. At what rate is the radius of the slick increasing when the radius is 8 m?

The volume of the oil slick is given by . Differentiate both sides with respect to time.

Volume is constant (1 ), so dV/dt=0. dh/dt is given as . , but we still need to find h at that time. r and h satisfy the volume formula, so use that formula to solve for h.

Now plug in all the unknown into the first equation.

After the necessary cancellations we get

which becomes

3) Use Newton's method to find the solution of

in the interval [11,12] accurate to four decimal places.

Start with and use the formula

We'd better find f' since we will use it in the formula.

Since the last two approximations have their first 4 decimal digits common, we conclude that the last approximation is accurate to 4 decimal digits.