Math 181

Test 3 will emphasize material from chapter 4, except for 4.5 which was skipped.

Some good odd-numbered problems to supplement the ones previously assigned are listed below. (There is a little overlap, but for the most part these are not the same problems)

4.1: Related Rates: 3, 11, 25

4.2: Maximum and minimum values: 13, 33, 37, 49

4.3: Mean Value Theorem (MVT), 1st and 2nd derivative tests. Also increasing/decreasing test and concavity test. 1, 13, 33, 41, 49

4.4: Using calculus and the calculator jointly to produce useful graphs: 1, 5

4.6: Optimization: 9, 13, 23, 37, 41, 45

4.7: Newton's Method: 7, 15, 31 (this last one is a good review of a couple of different topics we’ve covered!)

4.8: Antiderivatives: 3, 15, 33, 43

Here are some additional problems on which you can practice:

(Since 1 and 2 were discussed in class, here is another similar problem):

An object is dropped (released with 0 initial velocity) from a height of 400m. It undergoes an acceleration of

a(t)= -9.8+1.4t m/s^{2} 0≤t≤7, after which a(t) remains
constant at 0.

a. Find a formula for the velocity of the object during the first 7 seconds of its fall

b.
What is its velocity at time t=7 (and after, since the
acceleration is 0 from then on)? This is
called the object’s *terminal velocity*

c. How far does the object fall before reaching its terminal velocity?

1. The acceleration
due to gravity is -9.8 m/s^{2} .

Suppose a ball is thrown upward from a height of 2 m, with an initial velocity of 15 m/s.

a. Find a formula for the position of the ball as a function of time since it was thrown.

b. When does it reach its peak? How high does it get?

2. When the brakes
are applied to a car, it undergoes a uniform acceleration of -22 ft/sec^{2} (this is a
realistic braking force on a dry road – in slippery conditions, the
acceleration will not be as great).
Suppose you are driving a car at 60 mi/hr (=88ft/sec), and have to come
to a quick stop.

a. How long will it take you to come to a complete stop?

b. How far do you drive in that amount of time?

3. Use _{}. (What is a good
function to use for which this number will be a root? What would be a good initial guess?) Apply 2 steps of this method to get the next
2 improved estimates.

4. A computer monitor shows a picture of a “growing” rectangle. At a given time, it is 3 cm long and 2 cm high; the length is growing at a rate of 1 cm/sec, and the height is growing at a rate of 0.7 cm/sec. At what rate is the area growing?

5. Find the point on the curve _{} nearest to (5,0).

6. Maximize and minimize _{} subject to the
constraints x+y=5, x,y≥0

7. Find all critical points of f(x) = x + 25/x. Then find the absolute maximum and minimum values of this function on the domain [1,10]

8. F(x) is the antiderivative of the function shown in this

graph.Find
the critical numbers of F, and indicate whether

each is a local
maximum, a local minimum, or neither.

9.
You want to cut a rectangular beam from a cylindrical log of diameter 12
inches.

The
strength of a beam is proportional to the **product**
of the **width **and the **square of its depth**, so to make the
beam as strong as possible you want to maximize this product.

12 in

a) Express your objective as a function of 2 (or
more) variables.

b) What is/are your contraint
equation(s)

c) Solve for the exact
dimensions that will give the beam the greatest strength.