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Math 2053 Calculus I Highlights (Chapter 3) Definition of Extrema • Let f be defined on an interval I containing c. 1. f(c) is the minimum of f on I if f(c) ≤ f(x) for all x in I. 2. f(c) is the maximum of f on I if f(c) ≥ f(x) for all x in I. • The minimum and maximum of a function on an interval are the extreme values, or extrema (the singular form is extremum), of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute (or global) minimum or maximum on the interval. Definition of Extrema QuickTime ™ a nd a TIFF (U nc ompre sse d) dec ompre sso r are n eed ed to s ee th is picture. Definition of Relative Extrema • 1. If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f, or you can say that f has a relative maximum at (c, f(c)). • 2. If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f, or you can say that f has a relative minimum at (c, f(c)). Note: The plural of relative maximum (minimum) is relative maxima (minima). Critical Numbers • Let f be defined at c. If f ’(c) = 0 or if f is not differentiable at c, then c is a critical number of f. • If f has a relative minimum or relative maximum at x = c, then c is a critical number of f. • In other words, relative extrema of a function occur only at the critical numbers of the function. Critical Numbers a “cusp” QuickTime™ and a QuickTime™ an d a TIFF (Uncompressed) decompre ssor TIFF (U nco mpres sed ) de co mp ress or are neede d to see this picture. are ne ed ed to se e this pic tu re. Guidelines for Finding Extrema on a Closed Interval • To find the extrema of a continuous function f on a closed interval [a, b], use the following steps: 1. Find the critical number(s) of f in (a, b). 2. Evaluate f at each critical number in (a, b). 3. Evaluate f at each endpoint of [a, b]. I.e., Find f(a) and f(b). 4. The least of these values is the minimum. The greatest is the maximum. Rolle’s Theorem • Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b), then there is at least one number c in (a, b) such that f ’(c) = 0. Therefore, if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there must be at least one x-value between a & b at which the graph of f has a horizontal tangent. Rolle’s Theorem QuickTime™ an d a QuickTime ™ a nd a TIFF (U nco mpres sed ) de co mp ress or TIFF (U nc ompre sse d) dec ompre sso r are ne ed ed to se e this pic tu re. are n eed ed to s ee th is picture . The Mean Value Theorem • Rolle’s Theorem can be used to prove the Mean Value Theorem — and vice versa. Some consider these to be the most important theorems in calculus, along with the highly related Fundamental Theorem of Calculus. • If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f(b) – f(a) f ' c = b–a Increasing and Decreasing Functions • A function f is increasing on an interval if for any two numbers in the interval, implies that As x values increase, y values increase, and the graph moves up. • A function f is decreasing on an interval if for any two numbers in the interval, implies that As x values increase, y values decrease, and the graph moves down. • A function is said to be strictly monotonic on an interval if it is either increasing or decreasing on the entire interval. Derivative Test for Increasing and Decreasing Functions • Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). 1. If f’(x) > 0 for all x in (a, b), then f is increasing on [a, b]. 2. If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b]. 3. If f’(x) = 0 for all x in (a, b), then f is constant on [a, b]. Derivative Test for Increasing and Decreasing Functions QuickTime™ a nd a TIFF (U nc ompres sed ) d eco mpres sor are ne ed ed to se e this pictu re. Guidelines for Increasing and Decreasing Functions • Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing, use the following steps: 1. Locate the critical numbers of f in (a, b), and use these numbers to determine test intervals. 2. Determine the sign of f’(x) at one test value in each of the intervals. Guidelines for Increasing and Decreasing Functions 3. By the sign of f’, determine whether f is increasing or decreasing on each interval. • These guidelines are also valid if the interval (a, b) is replaced by an interval of the form –, b , a, , or –, . The First Derivative Test • Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then 1. If f’(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)). 2. If f’(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)). 3. If f’(x) is positive on both sides of c or negative on both sides of c, then f(c) is neither a relative minimum nor a relative maximum. The First Derivative Test QuickTime™ a nd a TIFF (U nc ompres se d) d eco mpres sor are ne ed ed to se e th is pictu re. Definition/Test for Concavity • Let f be differentiable on an open interval I. The graph of f is concave upward on I if f’ is increasing on the interval and concave downward on I if f’ is decreasing on the interval. • Let f be a function whose second derivative exists on an open interval I. 1. If f”(x) > 0 for all x in I, then the graph of f is concave upward in I. 2. If f”(x) < 0 for all x in I, then the graph of f is concave downward in I. Concavity QuickTime ™ a nd a Quick Time ™ and a TIFF (U nc ompre sse d) d ec ompres so r TIFF (Unc ompre ss ed) de comp ress or are n eed ed to s ee th is picture. are n eed ed to s ee this p ic ture . Concave up is like a “cup”; concave down is like a “crown”. Points of Inflection • Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c, f(c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point. • If (c, f(c)) is a point of inflection of the graph of f, then either f ”(c) = 0 or f ” does not exist at x = c. Points of Inflection Quick Time ™ a nd a QuickTime™ an d a TIFF (U nc ompre sse d) dec ompre ss or TIFF (U nco mpres sed ) de co mp ress or are n eed ed to s ee th is p icture . are ne ed ed to se e this pic tu re. Points of inflection can occur where f”(0) = 0 but (0,0) is not a point of f”(x) = 0 or f” does not exist. inflection The Second Derivative Test • Let f be a function such that f ’(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f ”(c) > 0, then f has a relative min at (c, f(c)). 2. If f ”(c) < 0, then f has a relative max at (c, f(c)). 3. If f ”(c) = 0, then the test fails. That is, f may have a relative maximum, a relative minimum, or neither. In this case, you can use the First Derivative Test. The Second Derivative Test QuickTime™ an d a Quick Time ™ a nd a TIFF (U nco mpres sed ) de co mp ress or TIFF (U nc ompre ss ed) de comp ress or are ne ed ed to se e this pic tu re. are n eed ed to s ee this p ic ture . Definition of a Horizontal Asymptote The line given by y = L is a horizontal asymptote of the graph of f if lim f(x) = L or lim f(x) = L x x– QuickTime ™ a nd a TIFF (U nc ompre sse d) dec ompre sso r are n eed ed to s ee th is picture . The graph of a function of x can have at most two horizontal asymptotes. Guidelines for Finding Limits at ± for Rational Functions • 1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0 (and the horizontal asymptote (H.A.) is y = 0). • 2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients (and the H.A. is y = ratio of leading coefficients). • 3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist (and there is no H.A.). Guidelines for Analyzing the Graph of a Function • 1. Determine the domain and range of the function. • 2. Determine the intercepts, asymptotes, and symmetry of the graph. • 3. Locate the x-values for which f ’(x) and f ”(x) either are zero or do not exist. Use the results to determine relative extrema and points of inflection. Optimization Problems 1. Identify all given quantities and quantities to be determined. If possible, make a sketch. 2. Write a primary equation for the quantity that is to be maximized or minimized. (Refer to geometry formulas inside the front cover of our text.) 3. Reduce the primary equation to one having . . . Optimization Problems 3. . . . a single independent variable. This may involve the use of secondary equations relating the independent variable of the primary equation. 4. Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense. 5. Determine the desired maximum or minimum value by using our calculus techniques. Volume of a Box Problem Find the “optimal” open box with a square base and a fixed surface area of 108 square inches. QuickTime ™ a nd a TIFF (U nc ompre sse d) dec ompre sso r In other words, find are n eed ed to s ee th is p icture . the dimensions of the box with maximum volume. Volume of a Box Problem These are a few of the QuickTime ™ a nd a TIFF (U nc ompre sse d) dec ompre sso r are n eed ed to s ee th is picture. possibilities! Newton’s Method Let f(c) = 0, where f is differentiable on an open interval containing c. Then, to approximate c, use the following steps: Make an initial estimate x1 that is close to c. (A graph is helpful.) Determine a new approximation using f x n xn + 1 = xn – f ' x n Newton’s Method (cont’d) f x n xn + 1 = xn – f ' x n If the value is within the desired accuracy, let xn+1 serve as the final approximation. Otherwise, return to step 2, and calculate a new approximation. Each successive application of this procedure is called an iteration. Newton’s Method QuickTime™ a nd a Quick Time ™ a nd a TIFF (U nc ompres sed ) d eco mpres sor TIFF (U nc ompre ss ed) de comp ress or are ne ed ed to se e this pictu re. are n eed ed to s ee this p ic ture . The x-intercept of the tangent line approximates the zero of f. Tangent Line Approximation Consider a function f that is differentiable at c. The equation for the tangent line at the point (c, f(c)) is given by y – f(c) = f ’(c)(x – c) QuickTime ™ a nd a TIFF (U nc ompres se d) d ec ompres so r are n eed ed to s ee th is picture. y = f(c) + f ’(c)(x – c) and is called the tangent line approximation (or linear approximation) of f at c. Differentials QuickTime™ a nd a TIFF (U nc ompres se d) d eco mpres sor are ne ed ed to se e th is pictu re. Differentials • Let y = f(x) represent a function that is differentiable on an open interval containing x. The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx. x = x – c y dy = f '(x) dx Error Propagation Formulas The main idea is using dy to approximate ∆y. ∆x represents the error in measurement (of height, radius, etc.); ∆y also represent the error in measurement (of area, circumference, volume, etc.) y dy - is called the relative error of the measurement y y Relative error is often converted to percent error. Differential Formulas • Let u and v be differentiable functions of x. Constant Multiple: d[cu] = c du Sum or Difference: d[u ± v] = du ± dv Product: d[uv] = u dv + v du u v du – u dv Quotient: d = v v2

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