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/s2  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/s2 .

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/sec2 (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 Newton’s Method to approximate the value of.  (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.

 


depth

 

   12 in

 

 

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

 

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

 

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