Dr. Lee - Math 35 - Calculus for Business - Review of 3 - Show Complete Work for Each Problem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find an equation of the line containing the pair of points. 9 ) 0, 5 7 and -9, 8 7 y = 0 23 x + 38 77 B) y = - 0 0 x C) y = - 23 23 x + 58 77 D) y = - 0 33 x + 38 2) A toilet manufacturer has decided to come out with a new and improved toilet. The fixed cost for the production of this new toilet line is $6,600 and the variable costs are $66 per toilet. The company expects to sell the toilets for $50. Formulate a function C(x) for the total cost of producing x new toilets and a function R(x) for the total revenue generated from the sales of x toilets. C(x) = 6,666; R(x) = 50 B) C(x) = 6600 + 50x; R(x) = 66x C) C(x) = 6600 + 66x; R(x) = 50x D) C(x) = 66x; R(x) = 50x 3) A construction company uses the function S(t) = 28,000-2000t to determine the salvage value S(t) of their trucks t years after it is purchases. What was the initial value of the truck and how long until it depreciates completely? $20000; years B) $28,000; years C) $28,000; 28 years D) $30,000; 5 years ) Suppose that the cost, p, of shipping a 3-pound parcel depends on the distance shipped, x, according to the function p(x) depicted in the graph. Find each limit, if it exists: lim x 00 p(x), x 500 lim p(x), x 500 lim p(x) 5; 0; 5 B) 5; 5; 5 C) 5; does not exist; 5 D) 5; does not exist; does not exist Find the limit, if it exists. 5) lim x -8 x2-6 x + 8 B) -6 C) Does not exist D) -8
6) lim x -7- x2-9 3.5 B) 7 7 C) 0 D) Does not exist Determine whether the function shown is continuous over the interval (-5, 5). 7) Yes B) No Evaluate or determine that the limit does not exist for each of the limits (a) for the given function f and number d. 8) f(x) = d = -, for x > -, x + x2-3x, for x - lim x d- f(x), (b) lim x d+ f(x), and (c) lim x d f(x) (a) Does not exist (b) 28 (c) 28 B) (a) 28 (b) Does not exist (c) 28 C) (a) 28 (b) Does not exist (c) Does not exist D) (a) Does not exist (b) 28 (c) Does not exist 2
Determine the continuity of the function at the given points. 2, for x = -, 9) f(x) = x - 2, for x - at x = - and x = 3 The function f is continuous at x = - but not at x = 3. B) The function f is continuous at both x = 3 and x = -. C) The function f is continuous at x = 3 but not at x = -. D) The function f is continuous at neither x = 3 nor x = -. Find the limit numerically. 36 + h - 36 -h 0) lim h 0 h 2 B) 0 C) 6 D) 6 ) Suppose that the dollar cost of producing x radios is c(x) = 200 + 0x - 0.2x2. Find the average cost per radio of producing the first 30 radios. $2.00 B) $20.00 C) $320.00 D) $.00 3
2) In one city, taxicabs charge passengers $2.00 for entering a cab and then $0.0 for each one-quarter of a mile (or fraction thereof) that the cab travels. (There are additional charges for slow traffic and idle times, but these are not considered here). If x is the distance traveled in miles, then C(x) is the cost of the taxi fare, where C(x) = $2.00, if x = 0, C(x) = $2.0, if 0 < x < 0.25, C(x) = $2.80, if 0.25 x < 0.5, C(x) = $3.20, if 0.5 x < 0.75, and so on. The graph of C is shown below. At what values is the function C not differentiable? 0.25, 0.5, 0.75,.0,.25,.5... B) Function is differentiable for all x in the domain C) 0.25, 0.5, 0.75 D) 0.25, 0.5, 0.75,.0 Find the simplfied difference quotient for the following function. 3) f(x) = 8 x + 2-8 (x + 2)2 B) -8 (x +h+2)(x + 2) C) 8 (x + 2)2 D) 8 (x +h+2)(x + 2) Find the derivative. ) f(x) = 8 x - 7 x + 5 x5 f'(x) = - x3/2-7 x2-25 x C) f'(x) = - x + 7 x2-25 x B) f'(x) = x/2-7 x2-25 x6 D) f'(x) = - x3/2 + 7 x2-25 x6 Find all values of x (if any) where the tangent line to the graph of the function is horizontal. 5) y = 3 x 3-2x + 7-2, 2 B) 7-2, 7 + 2 C) - 2, 2 D) -7, 7
6) Exposure to ionizing radiation is known to increase the incidence of cancer. One thousand laboratory rats are exposed to identical doses of ionizing radiation, and the incidence of cancer is recorded during subsequent days. The researchers find that the total number of rats that have developed cancer t months after the initial exposure is modeled by N(t) =.07t2.3 for 0 t 0 months. Find the rate of growth of the number of cancer cases at the 7th month. 26.2 cases/month B) 25.9 cases/month C) 30.9 cases/month D) 3.9 cases/month Differentiate. 7) f(x) = x + 5 x (x 2-6) [Do not use algebra before differentiating] f'(x) = 2x x + 5 x + - 5 x2 (x 2-6) B) f'(x) = 2x x + 5 x + ( - 5x)(x 2-6) C) f'(x) = x x + 5 x + + 5 x (x 2-6) D) f'(x) = 2x - 5 x2 8) h(r) = r 2 + 2r - 8 3r + 9 h'(r) = 3r 2 + 20r + 2 (3r + 9)2 C) h'(r) = 3r 2 + 8r + 2 (3r + 9)2 B) h'(r) = 3r 2 + 8r + 2 3r + 9 D) h'(r) = 2r + 2 3 9) g(x) = 5x + x + x2 8/5 g (x) = 8 5 5x + x + x2 C) g (x) = 8 5 5x + x + x2 20x 3 + - 8 x3 20x 3 + - 8 x B) g (x) = 8 5 5x + x + x2 D) g (x) = 8 5 20x 3 + - 8 x3 Find an expression for dy/. 20) y = u + u - and u = x + 9-8 x( x + 5) 2 B) ( x + 5) 2 C) 8 x( x + 5) 2 D) - x( x + 5) 2 Find the indicated derivative of the function. 2) d y of y = 3 8 80x/3 x B) - 8 80x/3 C) - 80 8x/3 D) 80 8x/3 5
Answer the question. 22) What information does the difference quotient, The instantaneous rate of change of f(x) as a function of x. B) The average rate of change of f(x) over the interval [x, x + h]. C) The limit of f(x) as x approaches h. D) The slope of the line tangent to f(x) at the point (x, f(x)). f(x + h) - f(x), provide about the differentiable function f(x)? h 23) What is the derivative of a function f(x)? The derivative of the function f(x) is a function, usually denoted f'(x), whose output f'(a) is the average rate of change of f(x) at the point (a, f(a)), where a is any value of x in the domain for f(x) where f'(x) exists. B) The derivative of the function f(x) is a function, usually denoted f'(x), whose output f'(a) is the average value of f(x) at the point (a, f(a)), where a is any value of x in the domain for f(x) where f'(x) exists. C) The derivative of the function f(x) is a function, usually denoted f'(x), whose output f'(a) is the instantaneous rate of change of f(x) at the point (a, f(a)), where a is any value of x in the domain for f(x) where f'(x) exists. D) The derivative of the function f(x) is a function, usually denoted f'(x), whose output f'(a) is the instantaneous value of f(x) at the point (a, f(a)), where a is any value of x in the domain for f(x) where f'(x) exists. 2) Suppose that y is a function of u, and that u is itself a function of x. How does one find the derivative of y in terms of x? The chain rule: dy = dy du du d(y + u) B) The sum rule: = dy + du C) The difference rule: d(y - u) = dy - du D) The product rule: d(y u) = y du + u dy Find the derivative. 25) y = e(8 x + x3) e( x + 3x2) B) (8 x + 3x2) e(8 x + x3) C) x + 3x 2 e(8 x + x3) D) (8 x + 3x2) ln (8 x + x3) 26) The nationwide attendance per day for a certain motion picture can be approximated using the equation A(t) = 2t2e-t, where A is the attendance per day in thousands of persons and t is the number of months since the release of the film. Find and interpret the rate of change of the daily attendance after months. 3.57 thousand persons/day month; the daily attendance is increasing. B) -.758 thousand persons/day month; the daily attendance is decreasing. C).758 thousand persons/day month; the change in the daily attendance is increasing. D) -3.57 thousand persons/day month; the change in daily attendance is decreasing. 6
Find the derivative of the function. 27) y = ln (8x3 - x2) 2x - 2 8x2 B) 2x - 2 8x2 - x C) 8x - 2 8x2 - x D) 2x - 2 8x3 - x 28) Suppose that the population of a town can be approximately modeled by the formula P = 2 ln where t is the time in years after 980 and P is the population of the town in thousands. Find an expression for dp/dt in terms of t. dp/dt = B) dp/dt = 6 6 ln C) dp/dt = D) dp/dt = 2 29) Ben Franklin bequeathed $000.00 to the city of Boston in 790. Assuming the fund grew to $ million in 200 years, find the interest rate compounded continuously that would yield this total value. 2.6% B) 5.7% C) 3.5% D).7% 30) A quantity Q grows exponentially with a tripling time of year. A quantity Q2 grows exponentially with a tripling time of 2 years. If the initial amounts of Q and Q2 are the same, when will Q be three times the size of Q2? after 3 years B) after.5 years C) after 2 years D) after 9 years 3) The supply and demand for the sale of television sets by an electronics company are given by S(p) = ln p and D(p) = ln 5,000, p where S(p) = the number of television sets that the company is willing to sell at $p and D(p) = the quantity that the public is willing to buy at $p. Find the equilibrium point. $392 B) $30 C) $372 D) $6 32) A business estimates that the salvage value V of a piece of machinery after t years is given by V(t) = $33,000e-0.9t. After what amount of time will the salvage value be $88? After 0.6 years B) After 9.6 years C) After 8.6 years D) After 7.6 years 7