APPM 1235 Final Exam Spring 2018

Transcription

1 APPM 35 Final Exam Spring 08 INSTRUCTIONS: Outside paper and electronic devices are not permitted. Exam is worth 50 points. Neatness counts. Unless indicated, answers with no supporting work may receive no credit. BOX your final answers.. ( points) The following are not necessarily related: (a) Find the di erence quotient, f(x + h) f(x) h, for the function: f(x) =x. (b) Solve the equation: ln x =+ln(x + ) (a) (x + h) x + h = x +xh + h x + h =x + h (b) ln(x + ) ln x = =) ln( x + x + )= =) = =) ex + e = x =) ex x = e =) x x e e x(e ) = e =) x = e = e e However, this makes x a negative value which makes ln x undefined. (8 points) Consider the function f(x) =4x 4 7x +4 (a) Use the Intermediate Value Theorem to demonstrate a root exists between x = 0 and x =. (b) Using arrow notation, describe the end behavior of f(x). (a) f(0) = 4 > 0 and f() = 9 < 0, therefore there exists 0 <a< such that f(a) =0. (b) x! =) f(x)!, and x! =) f(x)!

2 3. ( points) Consider the graph of f(x) =(x a)(x b)(x c), where a < 0 < b < c. (a) Sketch a graph of f(x). (b) What is the y-intercept of f(x)? (c) Where is f(x) < 0? Write your answer in interval notation. (a). (b) x = abc (c) (,a) [ (b, c) 4. ( points) The following are not necessarily related: (a) Given that tan = 5,findsec. (b) Sketch the graph of the equation y =sin x. (c) Find the exact value(s) of x in the interval [0, 4] that satisfy the equation: cos x = p (a). (b). sec = 3 (c) x = 4 +n and x = 7 4 x = 4, 7 4, 9 4, 5 4 +n

3 5. (8 points) Consider the circle below with the sector having central angle 45. (a) Find the length of the arc subtended by the 45 angle. (b) Find the area of the small sector. (a) S = r =) S =8 4 = (b) A = r = (8) 4 =8

4 6. (6 points) The pumping action of the heart consists of the systolic phase, in which blood rushes from the left ventricle into the aorta, and the diastolic phase, during which the heart muscle relaxes. The function whose graph is shown in the figure is sometimes used to model one complete cycle of this process with the equation y = a sin(bt). For a particular individual, the systolic phase lasts /4 second and has a maximum flow rate of 8 liters per minute. Find a and b. No work required to be shown. a =8,b =4

5 7. (36 points) The following are not necessarily related: (a) Business Calculus involves finding interest rates from an investment equation: A = P + i Solve this equation for i. 00. (b) When two electrical resistors with resistances R and R are connected in parallel, then the total resistance R is given by the expression: R = R +. If R = 0 ohms, and R = 0 ohms, then R what is the total resistance R? (c) Let V (d) be the volume of a spherical ball bearing of diameter d. To find the volume, one should take the cube of the diameter, then multiply by, and finally divide by 6. Find the volume of a sphere with radius. (d) In an electrical circuit the current I at time t is given by the formula I = 0e Rt L resistance and L is the inductance. Solve this equation for t. where R is the (e) The electrical resistance, R, of a wire varies directly as its length, l (in feet), and inversely as the square of its diameter (in inches), d. A wire 00 feet long of diameter 0. inch has a resistance of 5 ohms. Find the value of the constant of variation, k. (f) Find the horizontal asymptotes for the function: f(x) = x3 +4x 0 5x 3 x (a) A P = + i =) 00 r A P = i 00 r A =) i = 00 P 00 (b) R = (c) V = d = 3 0 = 0 3 =) V = 8 6 = 4 3 (d) I = 0e Rt L =) I 0 = e Rt L =) ln I 0 = Rt L =) t = L ln I 0 R (e) R = kl k 00 =) 5 = d =) k = 5 =) k = 5 0,000 =0.005 (f) x3 +4x 0 5x 3 x x 3 0 x 3 = x x 3 5! as x! 5 x

6 8. (36 points) The following questions are not necessarily related: (a) The number of bacteria in a certain culture is initially 500, and the culture doubles in size every day. Find a formula for the number of bacteria present after n days. (b) A pile of logs has 4 logs in the bottom layer, 3 in the second layer, in the third, and so on. The top layer contains 0 logs. Find the total number of logs in the pile. (c) =? (d) Write the following series in summation notation: (e) A curve, C, is described parametrically as x = t of C and indicate the orientation. and y =t + 3 for 0 apple t apple 5. Sketch the graph (f) Change the polar coordinates 4, 4, to rectangular coordinates. (a) a n = a + d(n ) = 500 n (b) S 5 = 5(0 + 4) = 5 34 = 5 7 = 55 (c) S = a r = + (d) (e). 7X n= n = 3 (f) x = r cos =) x = 4 cos( y = r sin =) y =4sin( 4 )=4 p = p 4 4 )=4 p = 4 p

7 9. (0 points) Answer each of the following; no work required: (a) What is the double-angle formula for sin(m)? (b) Factor: (a + b) + 5(a + b) 3 (c) Find the domain of the function: f(x) = x 4p 9 x (d) What is the addition formula for sin(m + N) =? (e) In terms of inequalities, write the statement: x is less than 3 and is greater than 5. (a) sin(m) =sin(m) cos(m) (b) (a +b )(a + b + 3) (c) ( 3, 3) (d) sin(m + N) =sin(m) cos(n)+sin(n) cos(m) (e) 5 <x< 3 Formulas that you may need: a n = a r n a n = a + d(n ) a n = a k r n k a n = a k + d(n k) S n = a ( r n ) r S = a r, r < S n = n(a + a n )