CALCULUS I 

REVIEW for FINAL EXAM

FALL 2001

1.      Given y1 = x3, y2 = x6 and y3 = x8  Sketch a graph (on the back of this page) which shows where the functions intersect and which are below/above the others where they do not intersect.

 

2.      Find the maximum or minimum point (whichever exists) and the x- and y-intercepts of f (x) = 2x2 - 6x + 8. (Bonus +1 if answer is obtained by completing the square.)

 

3.      Consider the graph of a quartic (4th degree) functio

a.       It has a minimum of ____ and a maximum of ____ x-intercepts.  Draw a freehand graph representing each possible case.

 

b.       It will always cross the - axis time(s).

 

4.      Sketch of graph showing how the tails of a polynomial function point depending on whether the degree is even or odd and on whether the leading coefficient is positive or negative.

a.        

.

Odd Degree 

Even degree 

Positive leading coefficient


 

 

.

Negative leading coefficient


 

 

.

5.      Give an example of a function which satisfies all of the following conditions:

a.       i) degree seven ii) odd symmetry iii) four terms Iv) left tail pointing upward

 

b.       Use the algebraic or graphical tests for symmetry to determine whether the graph of f(x) = x4 - x3 + 1 has odd, even or no symmetry.

 

6.       

a.       Fill in the chart below which shows how the degree of the polynomial in the numerator and denominator affect the horizontal or slant asymptotes.

Use >, <, = appropriately below:

HORIZONTAL OR SLANT ASYMPTOTE:

Degree of Num ___Degree of Den

.

Degree of Num ___Degree of Den

.

Degree of Num ___Degree of Den

.

b.       Find the vertical and horizontal asymptotes, and the x and y intercepts of .

 

7.      DERIVE identities for sin 2 x and cos 2 x , using the identity cos 2 x - sin 2 x = cos 2x and the Pythagorean identity to solve a pair of simultaneous equations.

 

8.      Fill in a unit circle with the sine and cosine values for the angles 0, /6, /4, /3, /2 and all other angles in [0,2) which have these angles as reference angles. (Use the back of this page.)

 

9.       Find the following limits.

a.      

b.      

c.      

d.       (Find the left, right and unsigned limit.)

 

10.  Find . Justify your answer using what you know about the behavior of polynomials as x .

 

11.  Where, if anywhere are the functions given below not continuous? If there are any x-values where the function is discontinuous, explain which part(s) of the definition of continuity is (are) true, and show why one part of the definition of continuity is false.

 

12.  State the definition of continuity for y = f (x) at x = a.

 

13.  State five definitions of the derivative.

 

14.  If the height of an object is given by the equation h(t) = 6t2 - 110t + 500, where h is in feet and t is in seconds, the average velocities over the intervals [10,11], [10,10.1], and [10,10.01] get successively and to the velocity at the time t = seconds. Explain your answer.

 

15.  Using the limit definition of the derivative find the derivative of y = x8.

 

16.  Find the derivative of . (Do not simplify)

 

17.  Find the derivative of . (Do not simplify)

 

18.  Find the slope of the line tangent to the graph of the equation 3y 4 + 4x2 - x2siny - 4x = 0, at the point (2,0).

 

19.  Approximate  using differentials. Compare this approximation to your calculator's value.

 

20.  Grain that is pouring from a chute at the rate of 8 cubic feet/min forms a conical pile whose altitude is always twice its radius. How fast is the altitude of the pile increasing at the instant when the pile is 6 ft high? 

 

21.   

22.  Choose two of the following derivations / proofs.  Others that you do are worth 2 points each.

a.       Derive the power rule.

 

b.       Derive the product rule.

 

c.       Prove that 

 

d.       Derive the derivative formula for f (x) = sin x

 

e.       Derive the chain rule

 

23.   

24.  Determine all of the following for the functions in problems 23 and 24:

§         The x- and y-intercepts, symmetry, and asymptotes;

§         A sign chart showing where the function

           Is increasing and decreasing,

           Where it has relative maxima and minima,

           Where it is concave up and down,

           And where it has inflection points;

§         Then sketch the graph by

           Plotting each key point and finding other necessary points,

           And finally by connecting these points with the correct shape using first and second derivative information

25.  f (x) =  

 

26.  g (x) = 

 

27.  Explain how to determine whether a function has a maximum or minimum at some x-value where the derivative is zero and also explain how to determine whether a function has an inflection point.

 

28.  A box with a square base is to have a volume of 4 cubic feet. Find the dimensions, which require the least amount of material.  (Disregard the thickness of the material and waste in construction.)  [Alternate problem: A hotel (with 60 rooms) charges $80 per day for a room and gives a special discount if more than 30 rooms are reserved. For each room over 30 rented out, the price of the room drops by $1. How many rooms must be rented for the hotel to maximize its daily revenue.]