Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval

Transcription

1 Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine. A critical number must be in the omain of the function! Relative Extrema: minimums or maximums of the entire function; function must be efine there f(x) is increasing when f (x) > 0 f(x) is ecreasing when f (x) < 0 First Derivative Test: Let c be a critical number for f(x). o If f (x) goes from positive to negative at x = c, f(c) is a relative maximum. o If f (x) goes from negative to positive at x = c, f(c) is a relative minimum. o If f (x) oes not change sign at x = c, f(c) is neither a relative min nor max. Notation: x = c is where the min/max occurs; f(c) is the value of the minimum or maximum. If they ask you to fin the minimum or maximum, they want you to fin the value of the function at that point. Lesson 19: Concavity, Inflection Points, an the Secon Derivative Test f(x) is concave up when f (x) > 0 f(x) is concave own when f (x) < 0 An inflection point is where f(x) changes concavity. To fin y-coorinate, plug into original function. Secon Derivative Test: Let c be a critical number for f(x) (i.e. f (c) = 0). o If f (c) > 0, f(c) is a relative minimum. o If f (c) < 0, f(c) is a relative maximum. o If f (c) = 0, the test is inconclusive, so use the first erivative test. Lesson 20: Absolute Extrema on an Interval How to fin the absolute extrema of f(x) on a close interval [a, b]: Fin all critical numbers of f(x). Evaluate f(x) at all critical numbers in [a, b], an at x = a an x = b. Compare the f(x) values an ientify the absolute min an max. Lesson 21: Graphical Interpretations of Derivatives When given a graph of f (x), we can fin where f(x) is/has the following: 1. critical numbers where f (x) = 0 or oes not exist, i.e. where f (x) touches the x-axis. 2. increasing - where f (x)>0, i.e where f (x) is above the x-axis.

2 3. ecreasing - where f (x)<0, i.e where f (x) is below the x-axis. 4. relative maximum - where f (x) goes from positive to negative. 5. relative minimum - where f (x) goes from negative to positive. 6. concave up - where f (x)>0, i.e, where f (x) is increasing 7. concave own - where f (x)<0, i.e, where f (x) is ecreasing 8. inflection point - f (x) changes sign, i.e. f (x) has a horizontal tangent line an switches from increasing to ecreasing or ecreasing to increasing Lesson 22: Limits at Infinity When evaluating a limit with x or x, take the highest x power in the numerator an enominator with their coefficients, cancel, an then evaluate. Asymptotes (for f(x)) o Vertical: Simplify f(x) an set the enominator equal to 0 an solve for x. (x = #) o Horizontal: Fin lim f(x) an lim f(x). If one or both exist an equal some finite L, y = L is the HA. x x o Slant: Only occur when the egree of numerator is exactly one more than the egree of the enominator. Fin by polynomial ivision. (Of the form y = mx + b.) Lesson 23: Curve Sketching To sketch a curve, fin the following: 1. x-intercepts (when y = 0) 2. y-intercept (when x = 0) 3. Increasing/Decreasing Intervals (use first erivative) 4. Concave Up/Concave Down Intervals (use secon erivative) 5. Inflection Points (use secon erivative) 6. Vertical Asymptotes 7. Horizontal Asymptotes 8. Slant Asymptotes When making number lines for the first an secon erivatives, be sure to mark where the function is unefine, check the sign (positive or negative) on both sies of the iscontinuity. None of the intervals shoul inclue the x-values where f(x) is unefine.

3 Lessons 24-26: Optimization Draw a picture! 1. Ientify quantity to be optimize (maximize or minimize). 2. Ientify the constraint(s). 3. Solve the constraint for one of the variables. Substitute this into the equation from Step 1. (If you have more than one constraint, you will have to be a bit more creative.) This is your objective function. 4. Take the erivative with respect to the variable. 5. Set the erivative equal to zero an solve for the variable. 6. Fin the esire quantity. Note: In most cases, you will only get one plausible solution. However, you shoul be sure on the exam to use the 1 st DT or 2 n DT to ensure what you foun is, inee, the minimum or the maximum. Lessons 27-28: Antierivatives an Inefinite Integration Rewrite the function, if necessary, to utilize the table of integration. We have yet to learn a metho to uno the chain, prouct, or quotient rules. Once you have integrate, you can take the erivative to ouble check your integration. When given an initial value or initial values, you can use them to fin the particular solution (i.e. you can solve for C).

4 Table of Derivatives x (c) = 0 Table of Integration 0 x = C x (xn ) = n x n 1, n 0 x n x = 1 n + 1 xn+1 + C, n 1 k x = kx + C x (ex ) = e x e x x = e x + C x (ln(x)) = 1 x, x > 0 1 x = ln x + C x (sin(x)) = cos (x) x (cos(x)) = sin (x) x x (tan(x)) = sec2 (x) (sec(x)) = sec(x) tan (x) x x (cot(x)) = csc2 (x) (csc(x)) = csc(x) cot (x) x (c f(x)) = c f (x) x x (f(x) ± g(x)) = f (x) ± g (x) x (f(x) g(x)) = f (x)g(x) + f(x)g (x) x (f(x) g(x) ) = f (x)g(x) f(x)g (x) (g(x)) 2 [f(g(x))] = f (g(x)) g (x) x x [ef(x) ] = f (x) e f(x) cos(x) x = sin(x) + C sin(x) x = cos(x) + C sec 2 (x) x = tan(x) + C sec(x) tan(x) x = sec(x) + C csc 2 (x) x = cot(x) + C csc(x) cot(x) x = csc(x) + C cf(x) x = c f(x) x (f(x) ± g(x)) x = f(x) x ± g(x) x We on t have a way to uno proucts or quotients except for the specific trig functions above. We also cannot uno the chain rule yet. To integrate proucts or erivatives, rewrite them!

5 Formulas to know: Shape Perimeter P or Surface Area S Area A or Volume V Rectangle with sies l an w P = 2l + 2w A = lw Square with sie x P = 4x A = x 2 Circle with raius r (Circumference C) C = 2πr A = πr 2 Triangle Trapezoi (with sies a, b, an c) P = a + b + c (with sies a, b, c, ) P = a + b + c + (with base b an height h) A = 1 2 bh (with base b 1 an b 2 an height h) A = 1 2 h (b 1 + b 2 ) Cube with sie x S = 6x 2 V = x 3 Rectangular prism with sies l, w, h S = 2lw + 2lh + 2wh V = lwh Revenue = units sol price Profit = units sol (price cost) Remember for optimization problems, rawing a picture will usually help you figure out what the equations are. In particular, pay attention to phrases like open top box or when a fence is boune on one sie. Also, if any picture is given, make sure you use the variables as they appear on the picture.