Math 113 HW #2 Solutions

Transcription

1 Math 3 HW # Solutions.3 : Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section., and then applying the appropriate transformations: y = x x + 3. Answer: We expect the graph of this function to be a parabola, translated from the origin so that the base of the parabola is centered on some point (a, b). The model for such a parabola is y = (x a) + b, which expands as If the given equation fits this model, then y = x a + (a + b). a = a + b = 3. Therefore, a = and b =. Hence, the given equation is equivalent to which is easy to graph: y = (x ), Figure : y = x x + 3

2 3: Find the functions (a) f g, (b) g f, (c) f f, and (d) g g and their domains, given Answer: f(x) = x, g(x) = 3 x. (a) (b) (c) (d) f g(x) = f(g(x)) = f ( 3 x ) = 3 x = 6 x has the domain {x R : x }. g f(x) = g(f(x)) = g ( x ) = 3 x has the domain {x R : x 0}. f f(x) = f(f(x)) = f ( x ) x = = x has the domain {x R : x 0}. g g(x) = g(g(x)) = g ( 3 x ) = 3 3 x has the domain R (i.e. there are no restrictions on the domain). : Express the function in the form f g. Answer: Define the functions x G(x) = 3 + x Then as desired. f g(x) = f(g(x)) = f f(x) = 3 x g(x) = x + x. ( ) x x = 3 + x + x = G(x), 56: An airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time t = 0. (a) Express the horizontal distance d (in miles) that the plane has flown as a function of t. Answer: Since the plane is traveling at a constant speed of 350 mi/h and since distance equals rate times time, we have that is the horizontal distance function. d(t) = 350t

3 d s Figure : Schematic picture of airplane and radar station.5 (b) Express the distance s between the plane and the radar station as a function of d. Answer: Using the Pythagorean Theorem, the distance s from the plane to the radar station is given by s = + d, so we have that s(d) = + d. (c) Use composition to express s as a function of t. Answer: If we want to express the distance from the radar station as a function of t, we simply take the composition s d(t) = s(d(t)) = s(350t) = + (350t) = +, 500t. 8: Find the exponential function f(x) = Ca x whose graph is given 0 ( ), 9 Answer: We know that the function f is of the form Ca x and that its graph passes through the two points (0, ) and (, /9). Since the graph passes through (0, ), we know that = f(0) = Ca 0 = C, 3

4 so we know C =. Since the graph passes through (, /9), we know that 9 = f() = a. Therefore, a = /9, meaning that a = /3. Therefore, the function f is given by ( ) x f(x) =. 3 0: Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? I. One million dollars at the end of the month. II. One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, n cents on the nth day. Answer: Option II. To see why, let s just figure out how much you would get paid on the last day of the month...and to make things as even as possible, let s assume the month is February (and not in a Leap Year). Since February has 8 days, on the last day of the month you would get paid 7 = 3,7,78 cents, which equals $,3,77.8. Therefore, under Option II, you get paid more on just the last day than you get paid for the whole month under Option I. 6: A bacterial culture starts with 500 bacteria and doubles in size every half-hour. (a) How many bacteria are there after 3 hours? Answer: Since 3 hours equals 6 half-hours, the culture will have doubled 6 times. Therefore, there will be = 3,000 bacteria. (b) How many bacteria are there after t hours? Answer: Since t hours is the same as t half-hours, the culture will have doubled t times. Therefore, there will be 500 t bacteria. (c) How many bacteria are there after 0 minutes? Answer: There are two possible answers depending on how you interpret the set-up to the problem. If each bacterium in the culture doubles once every half-hour on the half-hour, then each one will double after exactly 30 minutes, and then not again until 60 minutes have passed. In that case, there will be 500 = 000

5 bacteria after 0 minutes. On the other hand, if each bacterium doubles exactly once per half-hour, but at some random time within that half-hour, then it makes sense to think of the population function P (t) = 500 t as continuous. In that case, since 0 minutes is of an hour, the population will be after 0 minutes = = (d) Graph the population function and estimate the time for the population to reach 00, Answer: Using Mathematica or a graphing calculator, it s not too hard to check that the function P (t) = 500 t passes 00,000 at approximately t = : If you graph the function f(x) = e/x + e /x you ll see that f appears to be an odd function. Prove it Proof. We will prove the function f is odd by showing that f( x) = f(x). 5

6 To see this, consider Pull out a factor of e /x Therefore, e /x. e /x f( x) = e /x + e /x = + from both the numerator and denominator to get f( x) = ( e e /x ) /x ( e e /x + ) = e/x e /x +. /x f( x) = e/x = f(x), + e/x completing the proof that f is an odd function. 6