MATH 1325 Review for Test 3 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the quotient rule to find the derivative. 1 1) f(x) = x7 + 2 1) 2) y = x 2-3x + 2 x7-2 2) Find the derivative. (2x + 1)(2x + 2) 3) 2x - 5 3) 4) The total cost to produce x units of perfume is C(x) = (9x + 6)(9x + 8). Find the marginal average cost function. 4) 5) The total profit (in hundreds of dollars) from selling x items is given by P(x) = 8x - 5 4x + 5. Find the marginal average profit function. 5) Let f(x) = 8x2-5x and g(x) = 7x + 9. Find the composite. 6) f[g(3)] 6) 7) g[f(k)] 7) Find f[g(x)] and g[f(x)]. 8) f(x) = 5x + 9; g(x) = 4x - 7 8) 9) f(x) = x + 5; g(x) = 4x - 1 9) Find functions g and h such that f(x) = g(h(x)). 10) f(x) = (2x - 4)8 10) 11) f(x) = ln (4x5-5) 11) 12) f(x) = 1 5x + 2 12) 13) f(x) = e3x + 5 13) Find the derivative of the function. 14) y = (3x2 + 5x + 1)3/2 14) 1
x3 15) y = (x - 1) 3 15) 16) The total revenue from the sale of x stereos is given by R(x) = 3000 1 - average revenue from the sale of x stereos. x 2. Find the 200 17) $2300 is deposited in an account with an interest rate of r% per year, compounded monthly. At the end of 8 years, the balance in the account is given by A = 2300 1 + Find the rate of change of A with respect to r when r = 6. r 1200 96. 16) 17) Find the derivative. 18) y = e 6x2 + x 18) Find the derivative of the function. 19) y = ln (x - 7) 19) 20) y = x2 ln x2 20) 21) y = ln 9x2 21) Find the derivative. 22) f(x) = ln 9 + e 10x 22) 23) The sales in thousands of a new type of product are given by S(t) = 200-70e -.1t, where t represents time in years. Find the rate of change of sales at the time when t = 3. 24) Assume the total revenue from the sale of x items is given by R(x) = 31 ln (5x + 1), while the total cost to produce x items is C(x) = x. Find the approximate number of items that should 3 23) 24) be manufactured so that profit, R(x) - C(x), is maximum. Find all points where the function is discontinuous. 25) 25) 2
26) 26) Of the given values of x, identify those at which the function is continuous. 1 27) f(x) = ; x = 0, 9, -1 27) x(x - 9) 28) f(x) = x - 6 ; x = -6, 0, 6 28) x + 6 Write interval notation for the graph. 29) 29) Decide whether or not the function is continuous in the indicated x-interval. 30) -3 to 0 30) Find the location and value of each local extremum for the function. 31) 31) Identify the intervals where the function is changing as requested. 32) Increasing 32) 3
Find the largest open interval where the function is changing as requested. 33) Increasing y = 7x - 5 33) 34) Increasing f(x) = x 2-2x + 1 34) The graph of the derivative function f is given. Find the critical numbers of the function f. 35) 35) 36) 36) Determine the location of each local extremum of the function. 37) f(x) = x 3-5 2 x2 + 4x - 2 37) Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. 38) f(x) = (7-3x) 3/5-2 38) Solve each problem. 39) The price P of a certain computer system decreases immediately after its introduction and then increases. If the price P is estimated by the formula P = 160t 2-2100t + 6000, where t is the time in months from its introduction, find the time until the minimum price is reached. 39) 40) Find two numbers whose sum is 450 and whose product is as large as possible. 40) 4
Evaluate f''(c) at the point. 41) f(x) = 3x - 4 4x - 3, c = 1 41) 42) f(x) = (x 2-2)(x 3-5), c = 1 42) 43) Find the velocity function v(t) if s(t) = 3t 3 + 3t 2 + 9t - 5. 43) 44) Find the acceleration function a(t) if s(t) = -9t 3-3t 2 + 5t + 8. 44) Find the coordinates of the points of inflection for the function. 45) f(x) = x 3 + 33x 2 + 360x + 1301 45) Find the largest open intervals where the function is concave upward. 46) f(x) = 4x 3-45x 2 + 150x 46) Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. 47) f(x) = -x 3-4.5x 2 + 12x + 4 47) The rule of the derivative of a function f is given. Find the location of all local extrema. 48) f'(x) = (x + 4)(x + 1)(x - 4) 48) The rule of the derivative of a function f is given. Find the location of all points of inflection of the function f. 49) f'(x) = (x 2-1)(x - 5) 49) Find the location of the indicated absolute extrema for the function. 50) Maximum 50) 51) Minimum 51) 5
Find the absolute extremum within the specified domain. 52) Minimum of f(x) = x 3-3x 2 ; [- 0.5, 4] 52) 53) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where p(x) = 30 - x. How many candy bars must be sold to maximize 26 53) revenue? Find dy/dx by implicit differentiation. 54) x4/3 + y4/3 = 1 54) 55) x3 + 3x2y + y3 = 8 55) Find dy dx at the given point. 56) x2 + 3y2 = 13; (1, 2) 56) Find the equation of the tangent line at the given point on the curve. 57) x2 + y2 + 2y = 0; (0, -2) 57) 58) The demand equation for a certain product is 6p2 + q2 = 1500, where p is the price per unit in dollars and q is the number of units demanded. Find dq/dp. 58) Assume x and y are functions of t. Evaluate dy/dt. 59) xy + x = 12; dx = -3, x = 2, y = 5 59) dt 60) x3 + y3 = 9; dx dt = -3, x = 1, y = 2 60) 61) A company knows that unit cost C and unit revenue R from the production and sale of x R2 units are related by C = + 3127. Find the rate of change of revenue per unit when 282,000 the cost per unit is changing by $8 and the revenue is $1000. 61) 62) Given the revenue and cost functions R = 36x - 0.3x2 and C = 6x + 14, where x is the daily production, find the rate of change of profit with respect to time when 10 units are produced and the rate of change of production is 5 units per day. 62) 6
Sketch the graph and show all local extrema and inflection points. 63) f(x) = 2x 3-15x 2 + 24x 63) Evaluate. 64) 12x3 x dx 64) 65) (4x 11-7x 3 + 4) dx 65) 66) 8e 4y dy 66) Find the integral. 67) 9x -5-9x -1 dx 67) 68) (5x - 3) 2 dx 68) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 69) The slope to the tangent line of a curve is given by f'(x) = x 2-7x + 8. If the point (0, 4) is on the curve, find an equation of the curve. A) f(x) = 1 3 x3-7 2 x2 + 8x + 1 B) f(x) = 1 3 x3-7 2 x2 + 8x + 4 69) C) f(x) = 1 3 x3-8x 2 + 8x + 4 D) f(x) = 1 3 x3-8x 2 + 8x + 1 70) Find C(x) if C'(x) = 5x 2-7x + 4 and C(6) = 260. 70) A) C(x) = 5 3 x3-7 2 x2 + 4x + 2 B) C(x) = 5 3 x3-7 2 x2 + 4x - 260 C) C(x) = 5 3 x3-7 2 x2 + 4x - 2 D) C(x) = 5 3 x3-7 2 x2 + 4x + 260 7
Answer Key Testname: MATH1325 REVIEW FOR TEST 3 7x6 1) f'(x) = - (x7 + 2)2 2) y' = -5x 8 + 18x7-14x6-4x + 6 (x7-2)2 3) 8x2-40x - 34 (2x - 5) 2 4) 81-48 x2 5) -32x2 + 40x + 25 (4x 2 + 5x) 2 hundreds of dollars per item 6) 7050 7) 56k2-35k + 9 8) f[g(x)] = 20x - 26 g[f(x)] = 20x + 29 9) f[g(x)] = 2 x + 1 g[f(x)] = 4 x + 5-1 10) g(x) = x8, h(x) = 2x - 4 11) g(x) = ln x, h(x) = 4x5-5 12) g(x) = x-1/2, h(x) = 5x + 2 13) g(x) = ex, h(x) = 3x + 5 14) y' = 3 2 (6x + 5)(3x 2 + 5x + 1)1/2 3x2 15) y' = - (x - 1)4 16) R(x) = 3000 x 17) da dr = 295.52 18) 12xe 6x2 + 1 1 19) x - 7 20) 2x(1 + ln x2) 21) 2 x 10e10x 22) 9 + e10x 1 - x 2 200 23) 5.2 thousand per year 24) 93 items 25) x = -2, x = 0, x = 2 26) x = -2, x = 2 8
Answer Key Testname: MATH1325 REVIEW FOR TEST 3 27) -1 28) 0, 6 29) (-6, -2] 30) Continuous 31) (-3,-1), (-1,2), (2,1) 32) (-2, 2) 33) (-, ) 34) (1, ) 35) none 36) 0, 1 37) Local maximum at 1; local minimum at 4 38) Local minimum at 7 3, -2 39) 6.6 months 40) 225 and 225 41) f''(1) = -56 42) f''(1) = -2 43) v(t) = 9t 2 + 6t + 9 44) a(t) = -54t - 6 45) There are no points of inflection. 15 46) 4, 47) Local maximum at 1; local minimum at -4 48) Local maximum at -1; local minima at -4 and 4 49) -0.1, 3.43 50) (1, 3) 51) None 52) (2, -4) 53) 390 thousand candy bars 54) - x 1/3 y 55) - x 2 + 2xy x2 + y2 56) - 1 6 57) y = -2 58) dq/dp = -6p/q 59) 9 60) 3 4 61) $1128.00 62) $120.00 per day 9
Answer Key Testname: MATH1325 REVIEW FOR TEST 3 63) Local max: (1, 11) Local min: (4, -16) Inflection point: 5 2, - 5 2 64) 8 3 x9/2 + C 65) 1 3 x12-7 4 x4 + 4x + C 66) 2e 4y + C 67) - 9 4 x-4-9 ln x + C 68) 25 3 x3-15x 2 + 9x + C 69) B 70) A 10