SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Transcription

1 Math 1325 Ch.12 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the location and value of each relative extremum for the function. 1) 1) Identify the intervals where the function is changing as requested. 2) Increasing 2) Find the largest open interval where the function is changing as requested. 3) Decreasing f(x) = 4 - x 3) 4) Decreasing f(x) = x 3-4x 4) The graph given is the graph of the derivative of a function. Find the locations of all relative extrema of the function. 5) 5) y x Determine the location of each local extremum of the function. 6) f(x) = -x x x + 1 6) 1

2 7) f(x) = x x3 + 7x 2 + 8x + 2 7) Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. 8) f(x) = 4xe -x 8) Solve each problem. 9) If the price charged for a bolt is p cents, then x thousand bolts will be sold in a certain hardware store, where p = 64 - x 30. How many bolts must be sold to maximize revenue? 9) 10) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $3 per foot for two opposite sides, and $6 per foot for the other two sides. Find the dimensions of the field of area 600 ft 2 that would be the cheapest to enclose. 10) 11) Find two numbers whose sum is 280 and whose product is as large as possible. 11) Evaluate fʹʹ(c) at the point. 12) f(x) = 3x - 4 4x - 3, c = 1 12) 13) f(x) = ln (3x 2-2), c = -1 13) 14) Find the acceleration function a(t) if s(t) = -2 2t ) s is the distance (in ft) traveled in time t (in s) by a particle. Find the velocity and acceleration at the given time. 15) s = t 2-5, t = 3 15) Find the coordinates of the points of inflection for the function. 16) f(x) = 7e -x2 16) Find the largest open intervals where the function is concave upward. 17) f(x) = 4x 3-45x x 17) Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. 18) f(x) = -x 3-3x x ) 19) f(x) = x + 4 x 19) The rule of the derivative of a function f is given. Find the location of all local extrema. 20) fʹ(x) = (x 2-1)(x + 3) 20) 2

3 The rule of the derivative of a function f is given. Find the location of all points of inflection of the function f. 21) fʹ(x) = (x + 5)(x + 2)(x - 3) 21) 22) A truck burns fuel at the rate (gallons per hr) of G(x) = x + x while traveling at x 49 mph. If fuel costs $1.18 per gallon, find the speed that minimizes total cost for a 200-mile trip. Round to the nearest tenth. 22) 23) A company wishes to manufacture a box with a volume of 48 cubic feet that is open on top and is twice as long as it is wide. Find the width to the nearest foot of the box that can be produced using the minimum amount of material. 23) 24) A rectangular sheet of perimeter 39 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume? 24) Find the absolute extremum within the specified domain. 25) Maximum of f(x) = x 2-4; [-1, 2] 25) 26) Minimum of f(x) = (x 2 + 4) 2/3 ; [-2, 2] 26) 27) A piece of molding 185 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area? Round to the nearest hundredth if necessary. 27) 28) The velocity of a particle (in ft/s) is given by v = t 2-6t + 7, where t is the time (in seconds) for which it has traveled. Find the time at which the velocity is at a minimum. 28) Find dy/dx by implicit differentiation. 29) x3 + y3 = 5 29) 30) x3 + 3x2y + y3 = 8 30) 3

4 Find dy at the given point. dx 31) 4xe 7y = 19; (1, 0) 31) 32) 7x ln y = 16; (1, e) 32) Find the equation of the tangent line at the given point on the curve. 33) x2 + y2 = 25; (-4, 3) 33) 34) xy2 = 4; (4, 1) 34) Assume x and y are functions of t. Evaluate dy/dt. 35) x3 + y3 = 9; dx = -3, x = 1, y = 2 35) dt 36) A container, in the shape of an inverted right circular cone, has a radius of 6 inches at the top and a height of 10 inches. At the instant when the water in the container is 9 inches deep, the surface level is falling at the rate of -1.9 in./s. Find the rate at which water is being drained. 36) 37) One airplane is approaching an airport from the north at 148 km/hr. A second airplane approaches from the east at 199 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 36 km away from the airport and the westbound plane is 17 km from the airport. 37) Sketch the graph and show all local extrema and inflection points. 8x 38) f(x) = x ) Sketch a graph of a single function that has these properties. 39) (a) defined for all real numbers (b) increasing on (-3, -1) and (2, ) (c) decreasing on (-, -3) and (-1, 2) (d) concave upward on (-, -2) and (1, ) (e) concave downward on (-2, 1) (f) fʹ(-3) = fʹ(-1) = fʹ(2) = 0 (g) inflection point at (-2, 0) and (1, 1) 40) (a) defined for all real numbers (b) increasing on (-3, 3) (c) decreasing on (-, -3) and (3, ) (d) concave downward on (0, ) (e) concave upward on (-, 0) (f) fʹ(-3) = fʹ(3) = 0 (g) inflection point at (0, 0) 39) 40) 4

5 Answer Key Testname: M1325 CH 12 R FALL 16 1) (-3,-1), (-1,2), (2,1) 2) (-2, -1) (2, ) 3) (-, 4) 4) , ) rel. max.: -1 rel. min.: 1 6) Local maximum at 3; local minimum at -2 7) Local maximum at -2; local minima at -4 and -1 8) Local maximum at 1, 4 e 9) 960 thousand bolts 10) 34.6 $3 by 17.3 $6 11) 140 and ) fʹʹ(1) = ) fʹʹ(-1) = ) (2t + 2) 3 15) v = 1.5 ft/s, a = ft/s2 16) (-.71, 4.25) and (.71, 4.25) 15 17) 4, 18) Local maximum at 2; local minimum at -4 19) Local maximum at - 4; local minimum at 4 20) Local maximum at -1; local minima at -3 and 1 21) , ) 56.0 mph 23) 3 ft 24) x = 13 cm; y = 13 2 cm 25) (2, 0) 26) 0, ) cm x cm 28) 3 s 29) - x2 y 2 30) - x 2 + 2xy x2 + y2 31) ) - e 33) y = 4 3 x

6 Answer Key Testname: M1325 CH 12 R FALL 16 34) y = x ) ) in.3/s 37) 219 km/hr 38) Local minimum: (-4, -1) Local maximum: (4, 1) Inflection point: (0, 0), (-4 3, - 2 3), (4 3, 2 3) 8 y x -4 39) -8 40) 6