Solutions to Homework Assignment #2

Transcription

1 Solutions to Homework Assignment #. [4 marks] Evaluate each of the following limits. n i a lim n. b lim c lim d lim n i. sin πi n. a i n + b, where a and b are constants. n a There are ways to do this question. The first way is to use a special summation formula: n i lim n = lim n n n = 6. n i The other way is to first notice that lim n = lim i, since the last terms in this n latter sum clearly contribute only to the limit. Now recognize the sum in this last limit as a Riemann sum for b First note that lim hand side contribute lim d, and therefore lim i = lim i = d =. i since the missing terms on the left i 8 = lim =. Therefore n lim i = lim i = lim i =. c lim d lim πi sin n n = sinπ/ d = π cosπ/ = π. a in n + b = a + b d = a + b.

2 . [4 marks] Evaluate each of the following integrals. a f d, where fu =e u. b f d, wherefu =tanu. π/4 c d, where fu =sec u. d f π/6 e π/4 f + sinln d, where f = π/4 ln +ln. a We make the substitution u =, du = d: f d = e d = eu du = eu + C = e + C. b We make the substitution u =,du=d: π/4 f π/6 f d = tan d = tan udu = ln cos u + C = ln cos + C. c We make the substitution u =, du = d : π/4 sec d = d = π/6 = tanu = π/4 π/6 π/4 π/6 sec u du =tanπ/4 tanπ/6 d e π/4 = f f + eπ/4 + sinln d = e π/4 = fe π/4 f + = π 4 cos u π/4 e π/4 f d + e π/4 sinln sinln d by the Fundamental Theorem π/4 = π 4 +. d sin uduby the substitution u =ln, du = d

3 . [ marks] Derive the formula for the volume of a cone of radius r and height h by using the process of integration not the Fundamental Theorem of Calculus and then evaluating a limit. Putting the ais of the cone along the -ais from =to = h andthenusingtheright hand rule gives i=n V = lim π ri h n n = i=n i πr h lim n = πr h. See the diagram at the end. The net questions all use the Fundamental Theorem of Calculus in the form b a f d = fb fa. 4. [ marks] Suppose that a tank initially contained 5 gallons of water. Let V t denote the volume of water in the tank at time t. Suppose dv = k sinπt/6 in units of gallons dt per minute, where k is some constant. Determine k if the tank has 4 gallons after minutes. On the one hand V V = = 6k π V t dt = k cosπ/ = π, k sinπt/6 dt = 6k π cosπt/6 and on the other V V =. Comparing these answers we get k = π. 5. [ marks] The reaction time of a driver time it takes to notice and react to danger in the road ahead is about.5 seconds. When the brakes are applied, it then takes the car some time to decelerate and come to a full stop. If the deceleration rate is a = 8m/sec, how long would it take the driver to stop from an initial speed of km per hour? Include both reaction time and slowing-down time. First we must change units so all measurements are compatible, say km/hr = 6 m/sec = m/sec = 6 m/sec. Let vt denote the speed at time t and let T be the time the car comes to a full stop. Then vt v = T v t dt = T 8 dt = 8t T = 8T, but also vt v =. Therefore T = /4 = 5/6, and hence the time to stop is = 4 sec.

4 6. [ marks] The rate of flow of blood through the heart can be described approimately as a periodic function of the form F t =A + sin.5t, where t is time in seconds and A is a constant in units of cm /sec. Find the total volume of blood that flows through the heart between t =andt =. Epress your answer in terms of A. Let V t be the volume of blood in the heart at time t. Then V t =F t and so the volume of blood flowing through the heart between t =andt =isgivenby V V = V t dt = A + sin.5t dt = A.5 cos.5t = A cos = A.5 cos a. 7. [ marks] Find the area enclosed by y = f andy = g in the following cases: a f = and g = +. b f =e and g =/e. a The curves y = + and y = intersect at =and =. Since the line y = + is above the parabola y = in this range the area enclosed is A = + d = = = b The graphs intersect at = ± and are symmetric about the y-ais. Therefore the area is A = e d = e e e = 4 e. 8. [ marks] Use MathSheet for the following questions. Hand in the spread sheets. a Use the trapezoidal rule to approimate the area under the graph y = e from = to =, accurate to places. b Use the midpoint rule to approimate the length of the ellipse +y =, accurate to 4 places. a We first apply the trapezoidal rule to respective answers from MathSheet are: e d using n =,, 4 and n =6. The T = , T = , T 4 =.559, T 6 =

5 On this basis we begin to suspect that the answer to places is.5. To see that this is correct we could either compute the trapezoidal rule T n for larger and larger values of n or we could do some error analysis done in class. b The first quadrant part of the ellipse is given by y =,. Therefore the arc length of the total ellipse is given by L = 4 = 4 +y d =4 + + d = 4 d / d = 4 π/ sin θ dθ by the substitution =sinθ. Now we use the midpoint rule on this last integral. It turns out that we need only take n =4 to get 4 places of accuracy. M 4 =5.46. To verify this one must eperiment with larger values of n to see if increasing the value of n makes any difference. n a b c d e f Graph for question 7a.8.6 y Graph for question 7b 5