Download Review Sheet for Precalculus - Fall 2008 | MATH 3C and more Study notes Pre-Calculus in PDF only on Docsity! Math 3C Review Sheet UCSD, Winter 2008 1 Functions in general 1. Ways of expressing: table, graph, formula, words 2. Input and output; domain and range. Interval notation. 3. Describing change of a function: average rate of change, increasing and decreasing functions, concavity 4. Short run behavior: zeros, vertical asymptotes, holes, etc. 5. Long run behavior: horizontal asymptotes, dominance, long-run similarity of functions 6. Periodic functions: period, amplitude, midline. Trigonometric functions as examples. See below. 7. Piecewise defined functions. Example: absolute value function f(x) = |x|. 8. Operations on functions (a) Composition f(g(x)) (b) Arithmetic combinations f(x) + g(x), f(x)g(x) etc. (c) Inverse function f−1(y): find input for a given output i. Horizontal line test for invertible functions ii. Finding formula by solving y = f(x) for x iii. Graph: reflection about diagonal line y = x, domain and range swap iv. Composition of inverses: f(f−1(x)) = f−1(f(x)) = x v. Examples: see below. (d) Transformations of functions i. Inside and outside changes; horizontal and vertical ii. Shifts, stretches/compressions, reflections/flips. iii. Even and odd functions 1 Math 3C Review Sheet UCSD, Winter 2008 2 Specific families of functions 1. Linear functions (a) Constant rate of change (slope). Graph: straight line. (b) Formulas for linear functions: slope-intercept form y = mx+b, point- slope form y − y0 = m(x− x0), standard form Ax + By + C = 0. (c) Solving linear equations and linear systems. Parallel and perpendicu- lar lines. 2. Exponential and logarithmic functions (a) Properties of exponents and logarithms: log(ab) = log a + log b, etc. (b) Graphs and general shape. Example of inverse functions (c) Solving equations using/involving exponents and logarithms. Finding formulas for exponential functions. (d) Applications: interest, population growth, radioactive decay, logarith- mic scales, etc. 3. Trigonometric functions: sin t, cos t, tan t (a) Definitions in terms of unit circle. Special angles π 6 , π 4 , π 3 , π 2 , etc. Symmetry properties, even and odd functions. (b) Radian measure and arc length (c) Sinusoidal functions: transformations of sin t and cos t. Effect on pe- riod, amplitude, midline, phase shift. (d) Trigonometric identities: Pythagorean identity, double angle formula, phase shifts by 2π, π, π 2 . (e) Inverse trigonometric functions: arcsin t or sin−1 t, etc. Restriction of domain to avoid failure of horizontal line test. (f) Applications i. Solving right triangles: adjacent/opposite/hypotenuse ii. Solving non-right triangles: law of sines, law of cosines iii. Modeling periodic behavior 2
