Chapter 4 Review, pg. 1 Math 1a Review for Midterm II: Chapter 4 Course Assistant: Nicole Ali Dec. 11 2005 Section 4.1: Related Rates Notes: • Strategy: o Read the problem, draw a picture o Name the variables o Write down rates of change, known quantities o Write down an equation relating the variables o Differentiate with respect to time o Substitute known quantities o Solve for unknown variables • Helpful Formulae: o Area = ½ Bh = ½ ABsinө C2 = A2 + B2 – 2ABcosө ө A C h B Examples: From one of our handouts:Chapter 4 Review, pg. 2 Examples (Continued) Also from a previous exam: The Godzilla Problem: When Godzilla is standing at a 100 feet tall lamp-post, Godzilla is 10 feet tall. Godzilla then starts walking away from the lamp-post at a speed of 10 ft/sec, and at the same time he starts walking, Godzilla starts growing upwards at a rate of 1 f/sec. When Godzilla is 50 feet tall, what is the rate of change of the length of his shadow? ShadowChapter 4 Review, pg. 3 Section 4.2: Maximum and Minimum Values Notes: • f has an absolute maximum (global maximum) at c if f(c)≥f(x) for all x in the domain of f. • f has an absolute minimum (global minimum) at c if f(c)≤f(x) for all x in the domain of f. • The maximum and minimum values of f are called the extreme values of f • f has a local maximum (relative maximum) at c if f(c)≥f(x) when x is near c • f has a local minimum (relative minimum) at c if f(c)≤f(x) when x is near c • Note that an absolute max/min is sometimes also a local max/min • The Extreme Value Theorem o If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b] • Fermat’s Theorem o If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c) = 0 o Alternatively, if f has a local max or min at c, then c is a critical number of f • A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) doesn’t exist • The Closed Interval Method o To find the absolute max and min values of a continuous function f on a closed interval [a,b]:  Find the values of f at the critical numbers of f in (a,b)  Find the values of f at the endpoints of the intervals  See which values are largest/smallest Examples: 4.2.28: Find the critical numbers of the function g(t) = |3t – 4| 4.2.48: Find the absolute max and min values of f(x) = x – lnx on the interval [1/2, 2]Chapter 4 Review, pg. 4 Section 4.3: Derivatives and the Shapes of Curves f f’ f’’ Increasing Greater than 0 Decreasing Less than 0 Min/Max Equals 0 Concave up Increasing Greater than 0 Concave down Decreasing Less than 0 Notes: • The Mean Value Theorem o If f is differentiable on [a,b], then there exists a number c between a and b such that:  f’(c) = f(b) – f(a) b – a o The slope of the secant line between a and b is equal to the slope of the tangent line at some point c between a and b o The average velocity over an interval equals the instantaneous velocity at some point c • If f’(c) = 0, we use the first or second derivative test to see if c is a max, a min, or nothing o First Derivative Test:  If f’ changes from negative to positive, then c is at a min  If f’ changes from positive to negative, then c is at a max o Second Derivative Test:  If f’’(c) > 0, then f has a min at c  If f’’(c) < 0, then f has a max at c Examples: From a previous exam:Chapter 4 Review, pg. 5 Examples Continued: 4.3.54: Suppose that 3 ≤ f’(x) ≤ 5 for all values of x. Show that 18 ≤ f(8) – f(2) ≤ 30. From a Previous Exam: According to the intermediate value theorem, there is a point c in [π/2, π] such that f(c) = 0 4.4.57 Show that a cubic function always has exactly one point of inflection.Chapter 4 Review, pg. 6 Section 4.4: We Skipped This. Yay! Section 4.5: Indeterminate Forms and l’Hospital’s Rule Notes: • Indeterminate forms: 0/0, ∞/∞, 0·∞, ∞ - ∞, 00, ∞0, 1∞ • L’Hospital’s Rule (for type 0/0 or ∞/∞): o Suppose f and g are differentiable and g’(x) ≠ 0 near a (except possibly at a), and we have an indeterminate form of type 0/0 or ∞/∞. Then:  lim f(x) = lim f’(x) x→a g(x) x→a g’(x) • Dealing with type 0·∞ o Re-write the product as a quotient and use L’Hospital’s Rule • Dealing with type ∞ - ∞ o Try to re-write it as a quotient using rationalization, common denominator, or factoring • Dealing with type 00, ∞0, or 1∞ o Take the natural log of the limit L o Use L’Hospital’s rule to find ln L o Remember that L = elnL Examples: 4.5.46: Use L’hospital’s Rule to help fine the asymptotes of f(x) = ex/x From Previous Exams: Find the following limits.Chapter 4 Review, pg. 7 Section 4.6: Optimization Problems Notes: • Basic Strategy: o Draw a diagram o Introduce notation o Write down a function f for what you want to maximize or minimize o Write down the domain of f o Use calculus to find the absolute maximums or minimums (usually using the Closed Interval Method) • Remember the first and second derivative tests! • Remember to check the endpoints of your interval! Examples: From a Previous Exam: A numbers problem: Find two positive integers such that the sum of the first number and twice the second number is 30 and the product of the numbers is as large as possible. 4.6.30 A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 pm. At what time were the two boats closest together?Chapter 4 Review, pg. 8 Section 4.7: Applications to Business and Economics Notes: • Cost function: C(x) = cost of producing x units of a certain product • Marginal cost: C’(x) = rate of change of C with respect to x • Average cost function: c(x) = C(x)/x = cost per unit when x units is produced o If the average cost is minimum, then C’(x) = c(x) = C(x)/x • Demand function: p(x) = price per unit that the company can charge if it sells x units • Revenue function: R(x) = xp(x) = sales function = total revenue • Marginal revenue function: R’(x) •