Math - Some Algebra Review

This study guide is in no way exhaustive. As stated in class, any type of question from class, quizzes, exams, and homeworks are fair game. Every problem can have a variation or require multiple techniques. 1. Some Algebra Review (a) Logarithm Properties i. y = logb x is equivalent to x = by . ii. logb b = 1 iii. logb 1 = 0, ln 1 = 0 iv. logb (br ) = r, ln(er ) = r v. logb (xr ) = r logb (x), ln(xr ) = r ln(x) vi. logb (M N ) = logb (M ) + logb (N )   M vii. logb = logb (M ) − logb (N ) N viii. The domain of logb (u) is u > 0 ix. All the rules above hold for ln x. Remember that ln x = loge x x. Change of Base loga (M ) = logb M ln M = ln a logb a This is useful if you’re asked to find the derivative of y = log5 (x + 1) (b) Exponential Properties i. an am = an+m v. a−n = ii. (an )m = anm vi. iii. (ab)n = an bn an 1 iv. m = an−m = m−n a a 1 an 1 = an a−n vii. am/n = (am )1/n = (a1/n )m (c) Properties of Radicals i. √ n a = a1/n ii. √ n am = am/n 2. Limits (a) Notation i. General Limit Notation: lim f (x) = L x→a ii. Left Hand Limit: lim− f (x) = L x→a iii. Right Hand Limit: lim+ f (x) = L x→a iv. If lim− f (x) = lim+ f (x) = L then lim f (x) = L x→a x→a x→a (b) Limits at ±∞. Assume all polynomials are in descending order. 1 i. lim x→∞ a = 0, r > 0 xr iii. lim x→∞ ii. If n > m, then lim x→∞ axm + ... =0 bxn + ... axn + ... a = b bxn + ... iv. If m > n, then lim x→∞ axm + ... = ±∞ bxn + ... (c) Evaluation Techniques i. If f (x) is continuous, then lim f (x) = f (a) x→a lim x→3 ii. Factor and Cancel p x2 + 4 = If you evaluate a rational function and get lim x→3 p 32 + 4 = √ 13 0 , then try to factor. 0 x2 − 9 (x + 3) (x − 3)(x + 3) =6 = lim = lim x→3 x→3 1 x−3 x−3 iii. Rationalizing Numerators / Denominators Try this technique if you have radicals. Multiply top and bottom by the conjugate. lim √ √ 1 x − 16 x−4 x−4 x+4 √ = lim √ = lim = lim ·√ x→4 x − 16 x − 16 x + 4 x→4 (x − 16)( x + 4) x→4 x + 4 √ x→4 iv. Combine by Using Common Denominators Try this when you need to combine fractions within fractions lim x→a = lim x→a 2 x+4 2 − a+4 = lim x→a x−a 2 x+4 2 − a+4 (x + 4)(a + 4) 2(a + 4) − 2(x + 4) · = lim (x + 4)(a + 4) x→a (x − a)(x + 4)(a + 4) x−a −2(x − a) −2 −2 2a − 2x = lim = lim = x→a (x + 4)(a + 4) (a + 4)(a + 4) (x − a)(x + 4)(a + 4) x→a (x − a)(x + 4)(a + 4) (d) Piecewise Functions - Know how to graph and evaluate Piecewise Functions. These are good ones to test your understanding of left-hand, right-hand, and general limits. They are also used to test your understanding of continuity. (e) Definition of Continuity A function f is continuous at x = a if the following three conditions are satisfied: i. f (a) must exist ii. lim f (x) must exist x→a iii. lim f (x) = f (a) x→a 2 3. Derivatives (a) Limit Definition of a Derivative f 0 (x) = lim h→0 f (x + h) − f (x) h or f 0 (a) = lim x→a f (x) − f (a) x−a (b) Tangent Lines When asked to find the equation of a tangent line on f at x = a, you need two things: A point and a slope. i. You’re usually given the x value. If they don’t tell you the y value, you must plug the x value into f (x) to get the y value. Now you have a point (a, f (a)) ii. To find the slope m, you find m = f 0 (a). iii. The equation of the tangent line is y − f (a) = f 0 (a)(x − a). You may need to write this in slope-intercept form. (c) Derivative Formulas i. ii. iii. iv. v. vi. vii. viii. ix. x. xi. xii. xiii. xiv. d (c) = 0 dx d (f ± g) = f 0 ± g 0 dx d (x) = 1 dx d (kx) = k dx d (xn ) = nxn−1 Power Rule: dx d ([f (x)]n ) = n[f (x)]n−1 · f 0 (x) dx Product Rule: (f g)0 = f 0 g + f g0 f f 0 g − f g0 Quotient Rule: = g g2 0 Chain Rule: (f (g(x))) = f 0 (g (x)) · f 0 (x) d (ax ) = ax ln a dx d (ex ) = ex dx d 1 (ln x) = x dx 1 d (loga x) = dx x ln a d ef (x) = f 0 (x) · ef (x) dx xv. xvi. xvii. xviii. xix. xx. xxi. xxii. xxiii. xxiv. xxv. xxvi. xxvii. 1 d (ln(f (x))) = · f 0 (x) dx f (x) d (sin x) = cos x dx d (cos x) = − sin x dx d (tan x) = sec2 x dx d (sec x) = sec x tan x dx d (csc x) = − csc x cot x dx d (cot x) = − csc2 x dx 1 csc x = √ 2 dx x x −1 REMEMBER: all of these derivatives have a version for the chain rule. For example: 1 · 2e2x tan (e ) = 1 + (e2x )2 dx 3 (d) Critical Points x = c is a value of f (x) if f 0 (c) = 0 or f 0 (c) does not exist. (e) Increasing / Decreasing i. If f 0 (x) > 0 on an interval I, then f (x) is ii. If f 0 (x) < 0 on an interval I, then f (x) is increasing. decreasing. (f) Concave Up / Concave Down i. If f 00(x) > 0 on an interval I, then f (x) is ii. If f 00(x) < 0 on an interval I, then f (x) is concave up. concave down. (g) Inflection Points x = c is an inflection point of f if i. The point at x = c must exist. ii. Concavity changes at x = c (h) First Derivative Test to find Local (Relative) Extrema i. Find all critical value of f (x) where f 0 (c) = 0 ii. Local Max at x = c if f 0 (x) changes from (+) to (−) at x = c. iii. Local Min at x = c if f 0 (x) changes from (−) to (+) at x = c. iv. If f 0 (x) does not change signs, it’s still an important point to plot. It may be a place where the slope is 0, a corner, an asymptote, a vertical tangent line, etc. v. Make sure you write your relative max and mins as points (c, f (c)) (i) Second Derivative Test to find Local (Relative) Extrema i. Find all critical value of f (x) where f 0 (c) = 0 ii. Local Max at x = c if f 00 (c) < 0 iii. Local Min at x = c if f 00 (x) > 0 iv. Make sure you write your relative max and mins as points (c, f(c)) (j) Absolute Extrema i. (c, f(c)) is an absolute maximum of f (x) if f (c) ≥ f (x) for all x in the domain. ii. (c, f(c)) is an absolute minimum of f (x) if f (c) ≤ f (x) for all x in the domain. (k) Finding Absolute Extrema of a continuous f (x) over [a, b] i. Find all critical values of f (x) on [a, b]. ii. Evaluate f (x) at all critical values. iii. Evaluate f (x) at the endpoints, f (a) and f (b). 4 iv. The absolute max is the largest function value and the absolute min is the smallest function value.