Name: Class Period: Throughout this packet there will be blanks you are expected to fill in prior to coming to class. This packet follows your Larson Textbook. Do NOT throw away! Keep in 3 ring-binder until the end of the course. Definition of Extrema Chapter 3.1 Extrema on an Interval Extreme Value Theorem: If f is continuous on a closed interval [a,b], then f has both a maximum and a minimum on the interval. Definition of Relative Extrema Critical Numbers: 2 Conditions you must check to determine if x=c is a critical number: 1. 2. Theorem 3.2: Relative extrema occur only at Critical Numbers. So, if f has a relative max or min at x=c, then c is a of f. Question: If f =0 or does not exist at x=c, does that mean this is a relative max or min? 1 P a g e
Steps for finding Critical Numbers Step 1: Take f (x) Step 2: Find x-values where f (x)=0 (If you have a fraction set numerator =0) Step 3: Find x-values where f (x) does not exist (set denominator = 0) Step 4: Identify critical numbers: Critical Numbers: x=0, 1 Steps for Finding Extrema on Closed Interval Step 1: Find Critical Numbers in (a,b) Critical Numbers: x=0, 1 Step 2: Evaluate f at each critical number in (a,b) Step 3: Evaluate f at endpoints [a,b] Step 4: Make a chart. Least value is the minimum. Greatest value is maximum 2 P a g e
Rolle s Theorem 3.2 Rolle s Theorem and Mean Value Theorem To Use Rolle s Theorem: Let. Find all values in (-2,2) that have f (c)=0 Step 1: Show/state f(x) is continuous on [a,b] f(x) is polynomial, therefore continuous for all real x Step 2: Show/state f(x) is differentiable on (a,b) f(x) is polynomial, therefore differentiable for all real x Step 3: Show f(a) = f(b) So, f(-2) = f(2) Step 4: State there is a c in (a,b): f (c)=0 By Rolle s Theorem, there is at least one c in (-2,2) such that f (c)=0 Step 5: Solve f (x) = 0 for x values x = 0, 1, -1 The Mean Value Theorem Alternate form of Mean Value Theorem: Given find all value of c in the open interval (1,4) that satisfy the MVT Steps for using MVT: Step 1: Show/state f(x) is continuous on [a,b] f(x) continuous for all closed intervals that do not include x=0 Step 2: Show/state f(x) is differentiable on (a,b) f(x) differentiable for the open interval (1,4) Step 3: Evaluate f(a) and f(b) Step 4: Find f (x) Step 5: Write f (c) using MVT Step 6: Substitute in f, f values Step 7: Solve for x-value C = 2 is the value in (1,4) that satisfies MVT 3 P a g e
3.3 Increasing and Decreasing Functions and First Derivative Test Definition of Increasing and Decreasing Functions: Test for Increasing and Decreasing Functions Steps for finding Intervals on which f(x) is Increasing / Decreasing Find where the function on the interval (0, 2π) is increasing/decreasing Step 1: Show f is continuous on interval [a,b] f(x) is continuous for all x on [0, 2π] Step 2: Show f is differentiable on (a,b) f(x) is differentiable for all x on (0, 2π) Step 3: Find critical numbers of f in (a,b) Step 4: Make a sign chart using critical numbers Step 5: Pick one point in each interval to determine the sign of f Step 6: Use theorem to determine if f is increasing/decreasing Strictly Monotonic: 4 P a g e
1 st Derivative Test: Steps for Using 1 st Derivative Test to find max/min Find relative extrema of the function on the interval (0, 2π) Step 1: Show f is continuous on interval [a,b] f(x) is continuous for all x on [0, 2π] Step 2: Show f is differentiable on (a,b) f(x) is differentiable for all x on (0, 2π) Step 3: Find critical numbers of f in (a,b) ; Step 4: Make a sign chart using critical numbers Step 5: Pick one point in each interval to determine the sign of f Step 6: Identify Maximums (f changes from + to -) and Minimums (f changes from to +) 5 P a g e
3.4 Concavity and the Second Derivative Test Definition of Concavity Let f(x) be differentiable on an open interval I. The graph of f(x) is concave upward on I if f (x) is increasing on the interval and concave downward on I if f (x) is decreasing on the interval. Test for Concavity 1. If f (x) > 0 for all x in interval, then the graph of f(x) is 2. If f (x) < 0 for all x in interval, then the graph of f(x) is Definition of Points of Inflection Let f be a function that is continuous on an open interval. A point (c, f(c)) is an inflection point if the concavity changes at the point x=c Points of Inflection If (c, f(c)) is a point of inflection of the graph of f(x), then the either f (c)=0 OR f (c) does not exist. Steps for determining Concavity/Inflection Points Determine concavity of Step 1: Take 1 st Derivative: f (x) Step 2: Take 2 nd Derivative: f (x) Step 3: Set f (x)=0 Step 4: Make a sign chart for f (x) Step 5: Determine sign of f (x) Step 6: Use Test for Concavity to determine concavity Step 7: Identify inflection points where f changes sign. Inflection points: x = -1, 1 6 P a g e
The Second Derivative Test Steps for using 2 nd Derivative Test to find Maximums/Minimums Find relative extrema for using 2 nd Derivative Test Step 1: Find Critical Points Critical Points : Step 2: Take 2 nd Derivative : f (x) Step 3: Identify sign of f (x) at critical points Step 4: Use 2 nd Derivative Test to classify critical points Relative Minimum Test Fails Relative Maximum Step 5: If test fails, use 1 st Derivative Test Interval (-1, 0), f (x) > 0 so f(x) is increasing Interval (0,1), f (x) > 0 so f(x) is increasing By 1 st Derivative Test (0,0) is neither max nor min 7 P a g e
Definition of Limit at Infinity 3.5 Limits at Infinity Definition of Horizontal Asymptote Limits at Infinity: If r is a positive, rational number and c is any real number, then Guidelines for finding Limits at Infinity for Rational Functions 1. Degree of Numerator < Degree of Denominator, limit is zero 2. Degree of Numerator > Degree of Denominator, limit does not exist 3. Degree of Numerator = Degree of Denominator, limit is the ratio of the leading coefficients. Definition of Infinite Limits at Infinity 8 P a g e
3.6 A Summary of Curve Sketching Pre-Calculus x-intercepts and y-intercepts Set y=0, x=0 Symmetry even f(-x) = f(x), odd f(-x) = -f(x) Domain, Range Calculus Continuity Identify where f(x) is discontinuous Vertical Asymptotes Set denominator = zero Differentiability Where does f (x) not exist? Relative extrema f (x)=0, f (x) changes sign Concavity f (x) >0 or <0 Points of Inflection f (x)=0 and f (x) changes sign Horizontal Asymptotes Infinite Limits at Infinity Make a chart of intervals vs f(x), f (x) and f (x) and Characteristics Evaluate each point/interval Example: Sketch Domain: All real x Range -1 < Y < 1 Intercepts: (0,0) Vertical Asymptotes: None Horizontal Asymptotes: (+1 to right, -1 to left) No Critical Points Possible Inflection point at (0,0) Chart and Evaluation 9 P a g e
3.7 Optimization Problems Why???? You will be asked to Optimize (Maximize or Minimize) a variable that depends on other variables. Vocabulary: Primary Equation Equation that needs to be optimized Secondary Equations Equation that relate given quantities and known relationships. Steps for Optimization Problems: Step 1: Identify and write down any known measurements Step 2: Write down what you are looking for Step 3: Draw and label an accurate sketch Step 4: Identify Primary Equation Step 5: Identify Secondary Equations (usually is at least one) Step 6: Reduce Primary Equation to one variable using Secondary Equations Step 7: Identify reasonable domain Step 8: Take the derivative of Primary Equation Step 9: Find Critical Points Step 10: Check Critical Points and End Points 10 P a g e
3.8 Newton s Method Newton s Method uses tangent lines to approximate the graph of a function near its x-intercepts. Newton s Method for Approximating the Zeroes of a Function Steps for Using Newton s Method Step 1: Make a chart including n,,, Approximate a zero of, and. Use Step 2: Identify the initial point (x, f(x 1 )) Step 3: Find f (x) Step 4: Calculate f (x 1 ) Step 5: Calculate Step 6: Calculate Step 7: Record Step 8: Repeat Steps 4-7 for 2 nd Iteration Step 9: Repeat Step 8 until finished. 11 P a g e
Tangent Line Approximation: 3.9 Differentials Definition of Differentials Explain the difference in dy and Δy Error Propagation: Differential Formulas: Using Differentials to Approximate Function: Steps for Using Differentials to Approximate Approximate Step 1: Identify f(x), x and Let X = 16 and Step 2: Find f (x) Step 3: Approximate dx with Step 4: Write Step 5: Evaluate 12 P a g e