Math 121 Final Exam Notes

The final exam will be on Thursday Dec 7 at 9:00 a.m. in the GYM. As with the tests, you may use a non-programmable, non-graphing scientific calculator. You should also bring a ruler, some reliable pencils and an eraser. Scrap paper will be supplied if needed. You must also bring your Student ID with you as it will be checked during the exam.

No notes or formula sheets will be allowed; however, less familiar formulas (the surface area of a sphere for example) will be supplied if needed. You should know simple, standard formulas such as those for the area and perimeter of rectangles, circles and triangles, and also the volume of a box. You are expected to know the quadratic formula and the basic information on trig functions contained in useful facts & formulas.

The exam will consist of ten questions of a style similar to those of the three term tests. To get an idea of the style of the exam, take a look at the Math 121 exam from Summer 2006. The following is a partial list of focus topics:

• Limits and Continuity - Be able to calculate limits (including limits from left, right) and use limits to determine continuity of a function at a point.
• Definition of the Derivative - Know how to use the limit definition of the derivative to calculate the derivative of a function.
• Derivative Rules - There will be one or two questions which test your ability to apply the differentiation rules (+/-/×/÷/power/trig/chain). These may include some questions on computing antiderivatives and solving simple differential equations as well.
• Interpretations of the Derivative - Be familiar with the two interpretations of the derivative we have seen: f'(x) represents the slope of the tangent line to the graph of f(x), and also the rate of change of f(x) with respect to x. For the second of these, recall that if s(t)=displacement of an object at time t, then s'(t)=v(t)=velocity, and s''(t)=v'(t)=acceleration. You should also be able to determine the units for the derivative given the units for the original function.
• Tangent Lines & Approximation - You should know how to find the equation of the tangent line to a curve at a point, and how to then use this tangent line to approximate values the function near the point. These problems often require you to decide what the function being approximated is, and the point near which the approximation is being made. For example, if you are to approximate (9.1)^1/2 using a tangent line approximation, f(x)=x^1/2 while c=9. Recall how this concept was connected to differentials and error propagation.
• Implicit Differentiation - Be able to determine the first and second derivative of functions defined implicitly, and use the derivative to construct tangent lines to implicitly defined curves. In connection with tangent line approximation, these tangent lines may then be used to approximate the implicitly defined function near the point of tangency.
• Related Rates - You can expect a word problem on related rates.
• Optimization - You can expect one or two word problems on optimization. Recall the two classes of problems we saw:
  1. 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.
  2. 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.
• Curve Sketching - For a given function f(x), be able to
  1. determine intervals of increase/decrease and determine relative extrema using first derivatives;
  2. determine intervals where function is concave up/down and determine inflection points using second derivatives;
  3. determine horizontal and vertical asymptotes;
  4. determine x and y intercepts;
  5. put all of these items together to sketch the graph of f(x)
• Newton's Method - You should know how to implement Newton's Method to solve an equation of the form f(x)=0. In particular, be able to
  1. State the equation to be solved in the form f(x)=0. (This gives you f(x)).
  2. Determine the starting value x₁ using either a graph or table.
  3. Implement the method. For this you must know the formula x_n+1=x_n-f(x_n)/f'(x_n) (the formula will not be given).
  You should also know how Newton's Method is derived from the graph of the function the way we saw in class.