Preface — Advice for Students

The overwhelming emphasis in this course is on applications — story problems. To master story problems one needs a tremendous amount of practice — doing homework, worksheet, and practice exam problems. For this reason, this course will require much more of your time than would a high school or community college calculus course that does not have such a focus on word problems.

Here are some pointers for doing story problems:

1. Carefully read each problem twice before writing anything.

2. Assign letters to quantities that are described only in words; draw a diagram if appropriate.

3. Decide which letters are constants and which are variables. A letter stands for a constant if its value remains the same throughout the problem.

4. Using math notation, write down what you know and then write down what you want to find.

5. Decide what category of problem it is (this might be obvious if the problem comes at the end of a particular chapter, but will not necessarily be so obvious if it comes on an exam covering several chapters).

6. Double check each step as you go along; don’t wait until the end to check your work.

7. Use common sense; if an answer is out of the range of practical possibilities, then check your work to see where you went wrong.

Suggestions for Using These Notes

1. Read the example problems carefully, filling in any steps that are left out (ask someone if you can’t follow the solution to a worked example).

2. Later use the worked examples to study by covering the solutions, and seeing if you can solve the problems on your own.

3. Give yourself the practice midterms and finals in an exam situation (without access to material other than a single sheet of notes, and with the same time limit as on the exam).

4. The answers to homework and practice exam problems that are in the back of the notes should be used only as a final check on your work, not as a crutch. Keep in mind that sometimes an answer could be expressed in various ways that are algebraically equivalent, so don’t assume that your answer is wrong just because it doesn’t have exactly the same form as the answer in the back.

Formulas Used in Math 124 and 125

Algebra Completing the square: X^{2} + bX + c = (X + \frac{b}{2})^{2} − \frac{b^{2}}{4} + c . Quadratic formula: roots of aX^{2} + bX + c are \frac{−b ± \sqrt{b^{2} − 4ac}}{2a} .

Exponents: a^{b} · a^{c} = a^{b+c}; \frac{a^{b}}{a^{c}} = a^{b−c}; (a^{b})^{c} = a^{bc}; a^{1/b} = \sqrt[b]{a} .

Geometry Circle: circumference = 2πr; area = πr^{2} . Sphere: vol = \frac{4}{3}πr^{3}; surface area = 4πr^{2} . Cylinder: vol = πr^{2}h; lateral area = 2πrh; total surface area = 2πrh + 2πr^{2}. Cone: vol = \frac{1}{3}πr^{2}h; lateral area = πr\sqrt{r^{2} + h^{2}}; total surface area = πr\sqrt{r^{2} + h^{2}} + πr^{2}. Analytic geometry Point-slope formula for straight line: y = y_{0} + m(x − x_{0}). Circle centered at (h, k): (x − h)^{2} + (y − k)^{2} = r^{2}. Ellipse with semimajor axis along x-axis and semiminor axis along y-axis: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1. Trigonometry sin = \frac{opposite}{hypotenuse}; cos = \frac{adjacent}{hypotenuse}; tan = \frac{opposite}{adjacent}; sec = \frac{1}{cos}; csc = \frac{1}{sin}; cot = \frac{1}{tan}; tan = \frac{sin}{cos}; cot = \frac{cos}{sin}; sin x = cos (\frac{π}{2} − x); cos x = sin (\frac{π}{2} − x); sin(x + π) = − sin x; cos(x + π) = − cos x. Law of cosines: a^{2} = b^{2} + c^{2} − 2bc cos A. Law of sines: \frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} . Sine of sum of angles: sin(x + y) = sin x cos y + cos x sin y. sin^{2} and cos^{2} formulas: sin^{2} + cos^{2} = 1; sin^{2} x = \frac{1 − cos(2x)}{2}; cos^{2} x = \frac{1 + cos(2x)}{2}.