Math 121 Exam Notes

Exam Outline:

The exam will consist of approximately 12 questions, one question per page, some with multiple parts. You can expect

1. two pages of derivative calculations for which you must know your Table of Derivatives and Antiderivatives.
2. one page of limit calculations (possibly including left hand, right hand, limits involving infinity, and limits involving sin(x)/x as x approaches zero.) The concept of a continuous function as defined using limits may be included.
3. one page of antiderivatives.
4. one page involving the definition of the derivative and the interpretation of the derivative. Specifically,
   • for the definition you must know how to find derivatives using the limit definition.
   • you should be familiar with the two interpretations of the derivative we have seen:
     • f'(a) represents the slope of the line tangent to the graph of y=f(x) at the point (a,f(a)).
     • f'(x) represents the rate of change of f(x) with respect to x. Recall that if s(t)=displacement of an object at time t, then s'(t)=v(t)=velocity, and s''(t)=v'(t)=a(t)=acceleration. You should also be able to determine the units for the derivative given the units for the original function.
5. one page involving tangent line and linear approximation problems.
6. one related rate problem.
7. One or two optimization problems. Recall the two classes of problems we saw:
   • Find the absolute min/max of a continuous function on a closed interval. These involved testing the function at the critical numbers as well as the end points to determine the maximum and minimum values of the function.
   • Find the absolute min/max of a continuous function, possibly subject to a constraint. If a constraint is involved, the constraint is used to first reduce the objective function to a single variable. The function being optimized is then tested at the critical number(s) using either the first or second derivative test to determine the absolute min or max.
8. one or two pages dealing with derivatives and shapes of graphs. In particular, for a given function f(x), you should be able to
   • determine intervals of increase/decrease and determine relative extrema using first derivatives;
   • determine intervals where the function is concave up/down and determine inflection points using second derivatives;
   • determine horizontal and vertical asymptotes;
   • determine x and y intercepts;
   • put all of these items together to sketch the graph of f(x)
9. one page of implicit differentiation and logarithmic differentiation problems.
10. possibly one page combining several concepts in one problem.

The problems described above may involve any of the functions we have encountered during the term. As such, you must know your Table of Derivatives and Antiderivatives and also be familiar with the basic properties of the functions we have seen: polynomial, rational, root, trigonometric, exponential and logarithmic. You should know the graphs of y=sin(x), cos(x), tan(x), e^x, ln(x).

Extra Practice:

1. Final Exam Review Questions with answers (thanks to Dr.V. Watts for supplying these). I strongly urge you to work through these as they provide excellent preparation for the final exam. You may omit Questions 7, 18, 31-34, and please note the following typos in the answers:
   • The answer to 2(k) should be negative infinity
   • For Question 12(c), refer to the answer for 12(d) but replace 2x² with 2x
   • For Question 12(d), refer to the answer for 12(e)
   • For Question 12(e), refer to the answer for 12(c)
   • For the answer to 46(a) replace "1" with "0" in both intervals
2. Spring 2009 Final Exam (solutions).