Final Exam Review. b) lim. 3. Find the limit, if it exists. If the limit is infinite, indicate whether it is + or. [Sec. 2.

Transcription

1 Final Exam Review Math 42G 2x, x >. Graph f(x) = { 8 x, x Find the following limits. a) lim x f(x). Label at least four points. [Sec. 2.4, 2.] b) lim f(x) x + c) lim f(x) = Exist/DNE (Circle one) x 2, x 4 2. Graph f(x) = {. Label at least four points. [Sec. 2.4, 2.] 7 2x, x < 4 Find the following limits. a) lim f(x) b) lim f(x) c) lim x 4 x 4 + x x f(x) = Exist/DNE (Circle one). Find the limit, if it exists. If the limit is infinite, indicate whether it is + or. [Sec. 2.4] x a) lim 2 +4x+4 b) lim x2 8x+7 x(x c) lim 2 2) x 2 x 2 +x+2 x 7 x 2 6x 7 x 0 x 2 4 x d) lim e) lim 2 +x x 6 x 6 x 2x x 4. Use the definition of the derivative to find f (x). Show all the required steps. [Sec. 2.6] a) f(x) = 4x 2 x + 4 b) f(x) = 7 x 2. Find the derivative f (x). [Sec..] a) f(x) = x 4 x 2 + 6x b) f(x) = 2 2 c) f(x) = 2 4x d) f(x) = x 4 x 6. Find the derivative f (x) using the product rule of derivatives. Simplify, if possible. [Sec..2] a) f(x) = (x 2 + x)(x + ) b) f(x) = (x 2 + )(2 ) 7. Find the derivative y using the quotient rule of derivatives. Simplify, if possible. [Sec..2] a) y = x4 +x 2 + x b) y = x2 x+ 2x+ 8. Find the derivative f (x). Simplify, if possible. [Sec..] a) f(x) = b) f(x) = (4x 4x+) 2 x Evaluate the second derivative of the given function. [Sec..] a) f () for f(x) = x 2 b) f () for f(x) = 2 x

2 0. Solve the following applied problems. Interpret the results. [Sec..4] a) A steel mill finds that its cost function is C(x) = 8, dollars, where x is the daily production of steel (in tons). i. Find the Marginal Cost Function. ii. Find the marginal cost when 64 tons of steel are produced. iii. Interpret the results. b) The manufacturer determines that the profit P (in dollars) derived from selling x units is given by P(x) = x + 7x. i. Find the Marginal Profit Function. ii. Find the marginal profit for a production level of 2 units. iii. Interpret the results.. A company s profit function is P(x) = x,700. [Sec..4] a) Find the Relative Rate of Change of the Profit Function. b) Find the Marginal Profit Function. c) Evaluate the Relative Rate of Change of the Profit Function at x = Elasticity of Demand and applications. Elasticity of Demand: E(p) = pf (p). [Sec..4]. A South American country exports coffee and estimates the demand function to be f(p) = 6 2p 2. If the country wants to raise revenue to improve the balance of payments, should it raise or lower the prices from the present level of $ per pound?. Solve the following applied problems. Interpret the results. [Sec..] a) In a test flight of the McCord Terrier, McCord Aviation s experimental VTOL (vertical takeoff and landing) aircraft, it was determined that t seconds after liftoff, when the craft was operated in the vertical takeoff mode, its altitude (in feet) was h(t) = 6 t4 t + 4t 2, (0 t 8). i. Find an expression for the aircrafts velocity at time t. ii. Find the aircraft s velocity when t = 0, 2, and 8. iii. Find an expression for the aircraft s acceleration at time t. iv. Find the aircraft s acceleration when t = 0, 2. b) The median age (in years) of the U.S. population over the decades from 960 through 200 is given by f(t) = 0.276t +.962t 2 2.8t for (0 t ), where t is measured in decades, with t = 0 corresponding to 960. i. What was the median age of the population in the year 2000? ii. At what rate was the median age of the population changing in the year 2000? 4. Consider the function f(x) = 2x + x 2 + 2x + 2. [Sec. 4.] a) Determine the intervals where f is increasing and decreasing. b) Find the relative extrema of f. c) Determine the concavity of the graph of f. d) Find the inflection points of f. e) Plot a few additional points and sketch the graph of f. f(p)

3 . Find the absolute extreme values for each function on the given interval. [Sec. 4.4] a) f(x) = x 2x 2 4x + 2 on [,]. b) f(x) = x + 6x 2 + 9x + on [ 4,2]. 6. Solve the following optimization problems. [Sec. 4.] a) The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 26 cubic inches. The material for the top costs $0.0 per square inch and the material for the sides and bottoms costs $0.0 per square inch. Find the dimensions of the box that will minimize the cost. b) A farmer wants to make three identical rectangular enclosures along a straight irrigation ditch. If he has,200 yards of fence, what would be the dimensions of each enclosure if the area is to be maximized and the ditch needs no fence? c) A company manufactures and sells x videophones per week. The weekly price-demand and cost function are respectively p(x) = 00 0.x and C(x) = 20, x What is the maximum weekly profit? How much should the company charge for the phones and how many phones should be produced to obtain the maximum weekly profit? 7. Differentiation of Exponential Function. Find the derivative of the function. [Sec..4] a) y = 6e x2 b) y = x 2 e x c) y = ln(e 4x + 2) 8. Differentiation of Natural Log. Find the derivative of the function. [Sec..] a) y = ln x ln (x ) b) y = ln x4 x 2 +9 c) y = ln 2 9. Find the integral. [Sec. 6.] a) dx b) (x 4 9x 2 + ) dx c) (x x6) dx 20. Integration using logarithmic and exponential functions. [Sec. 6.2] a) 8 e 4x dx b) (e 4x x 2x dx c) dx x) x 2 2. Find the particular solution y = f(x) that satisfies the differential equation and initial condition. [Sec. 6.2] a) Suppose that the marginal revenue from the sale of x units of the product is MR = R = 6e 0.0x. What is the revenue in dollars from the sale of 00 units of the product? b) The rate at which blood pressure decreases in the aorta of a normal adult after a heartbeat is dp = dt 46.64e 0.49t, where t is the time in seconds. i. What function describes the blood pressure in the aorta if p = 9 when t = 0? ii. What is the blood pressure 0. seconds after a heartbeat? e2

4 22. Evaluate the definite integral. [Sec. 6.] 0 a) 4 2 dx 0 b) e x dx e c) 4 z dz 2. Find the points of intersection and the bounded area between the two curves. [Sec. 6.6] a) y = x 2 and y = x + 2 b) y = x x 2 and y = x 2 4x Key:. a) b) c) DNE 2. a) - b) 2 c) DNE. a) 0 b) ¾ c) + d) /8 e) 0 4. a) 8x- b) -2x. a) 20x 2 6x + 6 b) c) d) x x 6. a) 9x 2 + 8x + b) x x(2 ) 7. a) 2x (x 2 +2) b) 2x2 +0x 7 (x 2 +) 2 (2x+) 2 8. a) (2x2 4) b) 2x (4x 4x+) 4 (x 2 x+4) 2/ x / 9. a) -4/ b) -9/4 0a. i) C (x) = ii) 487. x2/ iii) The cost is increasing $487. for every ton of steel when the mill is producing 64 tons of steel. 0b. i) P (x) = 0.002x ii) 0.24 iii) The profit is increasing at $0.24 per unit when the manufacturer produces 2 units.. a) P (x) P(x) x 700 b) P (x) = c) E() = 4 ; Raise price a. i) v(t) = 4 t t 2 + 8t i) v(0) = 0; v(2) = 6; v(8) = 0 iii) a(t) = 4 t2 6t + 8 iv) a(0) = 8; a(2) = b. i) f(4) =., Median Age (years) ii) f (4) = a) Increasing: (,2); Decreasing: (, ) (2, ) b) Rel. Max: (2,22) and Rel. Min: (-,-) c) Concave up: (, ) and Concave down: 2 (, ) 2 d) Inflection Point: ( 2, 8.) e). Min: 6 at x=2; Max:.48 at x = 2 6. a) 4in x 8in x 8 in b) 200 yd x 0 yd b) Min: (-4,-) & (,-); Max: (2,)

5 c) Max profit is $46,62.0 when 6 phones are produced and sold for $7.0 each. 7. a) 6xe x2 b) xe x (x + 2) c) 4e4x 8. a) x x b) 4 x 2x x a) a) 2e 4x + C b) e4x c) e 4x +2 2x (x 2 ) + C b) x x + x + C c) x4 4 4x x + C e + C c) ln x x2 + C 2 x 2. a) $,00.97 b) i) p = 9e 0.49t ii) a) (2) 2/.088 b) e 6.62 c) 4 2. a) 9/2 b) 2/24.208