Math 265 Test 3 Review
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1 Name: Class: Date: ID: A Math 265 Test 3 Review. Find the critical number(s), if any, of the function f (x) = e x 2 x. 2. Find the absolute maximum and absolute minimum values, if any, of the function f (x) = x x on [0, 25] 3. Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. f( x) = 2 x, [ 0,25] 4. The function f (x) = 3 x 3 6x satisfies the hypotheses of Rolle s Theorem on the interval È 0, 4 3 ÎÍ. Find all values of c that satisfy the conclusion of the theorem. 5. Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. f( x) = x 3 5x 2 + 6x + 2, [ 0, 4] 6. Determine where the graph of the function f (x) = 9x 4 x 2 is concave upward and where it is concave downward. Also, find all inflection points of the function. 7. How many points of inflection are on the graph of the function? f( x) = 8x 3 + 5x 2 2x Find the inflection points for the function given. f( x) = 5x + 2 sinx, 0 < x < 3π 9. Find the critical numbers of the function. y = x x Find the limit. lim t 0 4 t 3 t t
2 Name: ID: A. The graph of the derivative f ( x) of a continuous function f is shown. On what intervals is f decreasing?. 2. Find the limit. 3 lim 5 cos(x) ( ) x π / 2 3. Find the limit. lim x x 2 2 5x Sketch the curve. y = x x 5. Find the limit. lim x 0 x tan ( 2x) 2
3 Name: ID: A 6. Sketch the curve. Find the equation of the slant asymptote. y = x 2 x 7. The quantity demanded per month of an item is related to the unit price by the demand equation p = x 2 + 5, 0 x 20 where p is measured in dollars and x is measured in units of a thousand. How many items must be sold by the manufacturer to maximize its revenue? Hint: Recall that the revenue is given by R = px. 8. The graph of the first derivative f ( x) of a function f is shown below. At what values of x does f have a local maximum or minimum? 9. Sketch the graph of the function f (x) = 3x 2 x 2 + using the curve-sketching guidelines. 20. Find the critical number(s), if any of the function f (t) = 2t 4 4t 2. Find the critical number(s), if any, of the function h (u) = u u
4 Name: ID: A 22. A manufacturer has been selling,200 television sets a week at $400 each. A market survey indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function. 23. The owner of a ranch has 4000 yd of fencing with which to enclose a rectangular piece of grazing land situated along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area? 24. A production editor decided that a promotional flyer should have a -in. margin at the top and the bottom, and a -in. margin on each side. The editor further stipulated that the flyer should have an 2 area of 392 in. 2. Determine the dimensions of the flyer that will result in the maximum printed area on the flyer. 25. Sketch the graph of the function g (x) = x 2 x using the curve-sketching guidelines. 26. A piece of wire 0 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut for the square so that the total area enclosed is a minimum? Round your answer to the nearest hundredth. 27. A rectangular storage container with an open top is to have a volume of 0 m 3. The length of its base is twice the width. Material for the base costs $2 per square meter. Material for the sides costs $5 per square meter. Find the cost of materials for the cheapest such container. 4
5 Name: ID: A 28. Find two positive numbers whose product is 2 and whose sum is a minimum. 29. Find the slant asymptote of the function f (x) = x x 30. Find the absolute maximum value of y = 6 x 2 on the interval [ 6, 6] Estimate the value of by using three iterations of Newton s method to solve the equation x 3 = 0 with initial estimate x 0 = 2. Round your final estimate to four decimal places. 32. Use Newton's method with the specified initial approximation x to find x 3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) x 4 2 = 0, x = Use Newton's method to approximate the indicated root of x 4 + x 4 = 0 in the interval [,2], correct to six decimal places. Use x =.5 as the initial approximation. 5
6 Math 265 Test 3 Review Answer Section. ANS: 2 PTS: DIF: Medium REF: MSC: Bimodal 2. ANS: Abs. max. f (25) = 20 Ê ˆ abs. min. f Ë Á 4 = 4 PTS: DIF: Medium REF: MSC: Bimodal 3. ANS: c = 25 4 PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: 4 PTS: DIF: Easy REF: MSC: Bimodal NOT: Section ANS: c = , c = PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: CU on ( 2, 2), IP none PTS: DIF: Difficult REF: c MSC: Bimodal NOT: Section 4.3
7 7. ANS: PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: ( π,5π + 2), ( 2π, 0π + 2) PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: 6, 6 PTS: DIF: Medium REF: MSC: Numerical Response 0. ANS: ln4 ln3 PTS: DIF: Medium REF: MSC: Bimodal NOT: Section 4.4. ANS: ( 4,6) ( 7, 9) PTS: DIF: Medium REF: 4.3.3a MSC: Bimodal NOT: Section ANS: PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: 5 PTS: DIF: Easy REF: MSC: Bimodal NOT: Section 4.4 2
8 4. ANS: PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: 2 PTS: DIF: Medium REF: MSC: Numerical Response NOT: Section ANS: y = x + PTS: DIF: Medium REF: MSC: Numerical Response NOT: Section 4.5 3
9 7. ANS: 0,000 PTS: DIF: Difficult REF: MSC: Short Answer 8. ANS: 3, 5, 8, 0 PTS: DIF: Medium REF: 4.3.3b MSC: Numerical Response NOT: Section ANS: PTS: DIF: Medium REF: MSC: Short Answer NOT: Section ANS: 0, 4 9 PTS: DIF: Medium REF: MSC: Bimodal 2. ANS: ± 6 PTS: DIF: Medium REF: MSC: Bimodal 22. ANS: p( x) = 0.5x +,000 PTS: DIF: Medium REF: MSC: Bimodal 4
10 23. ANS: Dimensions: 2000 yd 000 yd, Maximum area: 2,000,000 yd 2 PTS: DIF: Medium REF: MSC: Short Answer 24. ANS: 4 in. 28 in. PTS: DIF: Difficult REF: MSC: Bimodal 25. ANS: PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: 4.35 m PTS: DIF: Medium REF: MSC: Bimodal 27. ANS: $53.9 PTS: DIF: Medium REF: MSC: Bimodal 28. ANS:, PTS: DIF: Medium REF: MSC: Bimodal 5
11 29. ANS: y = x PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: 4 PTS: DIF: Medium REF: MSC: Bimodal 3. ANS: PTS: DIF: Medium REF: MSC: Bimodal NOT: Section ANS: PTS: DIF: Medium REF: MSC: Numerical Response NOT: Section ANS: PTS: DIF: Medium REF: MSC: Numerical Response NOT: Section 4.8 6
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