# Math 121: Final Exam Study Guide

1. A graph of a function is given.
1. Find the limits graphically.
2. Find the values of the function.
3. Determine continuity.
4. Determine where the derivative exists.
5. Make sign charts for the function and its first two derivatives.
2. Find the limits algebraically. Show work.
1. Limits are of the form 0/0, ∞/∞, or ∞-∞
2. Know the three basic trigonometric limits: $$\mathop {\lim }\limits_{t \to 0} \left( {{{\sin t} \over t}} \right) = 1$$, $$\mathop {\lim }\limits_{t \to 0} \left( {{{\tan t} \over t}} \right) = 1$$, and $$\mathop {\lim }\limits_{t \to 0} \left( {{{1 - \cos t} \over t}} \right) = 0$$
3. Know the second fundamental theorem of calculus to find the derivative of an integral.
4. Make a u-substitution and rewrite an integral completely in terms of u. Do not evaluate the integral.
5. Solve the initial value problem.
6. Find the derivative.
1. Know the power, product, quotient, and chain rules.
2. Know the derivatives of polynomial, rational, trigonometric, logarithmic, and exponential functions.
3. Know how to differentiate implicitly.
7. A function and its first two derivatives are given in factored form.
1. Identify whether the function is increasing and decreasing and where it is concave up and concave down.
2. Identify all critical points and whether they are relative maximums, relative minimums, or neither.
3. Identify all points of inflection.
8. Given the value of a function and its derivative, find a linear approximation for the function at a nearby point.
9. A curve and the area between the curve and the x-axis on selected intervals is given. Use it to find the indicated integrals.
10. Integrate.
1. Know the basic trigonometric integrals.
2. Know how to use u-substitutions.
3. Be able to find definite and indefinite integrals.
11. A region between two curves is given. Write definite integrals and then use the numeric integration feature of your calculator to find the following:
1. the area between the curves
2. the volume of the solid when the region is rotated about an axis
3. the area of the surface generated when the region is rotated about an axis
4. the length of a curve
5. the centroid of the region

## Notes

• You may use a calculator, however, give exact answers on the exam. Decimal approximations will result in reduced points.
• Be sure to use the proper terminology: A limit exists or does not exist. Functions are defined or undefined.
• Show work when necessary and also when asked to show work.
• You will need paper for scratch work. Write answers on the exam.
• Write neatly and legibly.
• Use proper notation when finding limits, integrals, and derivatives. Points will be deducted for improper notation.
• You will need a working knowledge of trigonometry for this exam. This includes the values of the trigonometric functions for the common angles. The derivatives of the six functions and the integrals that result in the basic functions. You do not need the sum/difference of two angles, double angle, or half-angle formulas.
• The points below add to 214, but the maximum you may obtain is 200 points. This means you may miss up to 14 points without affecting your grade.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 Total 32 36 8 6 8 36 16 6 12 30 24 214