REVIEW FOR PRELIM 1 SOLUTIONS

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1 REVIEW FOR PRELIM 1 SOLUTIONS MATH 1106, Spring 2017 Tutorial 9 Solutions Wednesday 02/22/17 Tutorial 9.1. Linear approximation (a) Let C(x) = 3x (i) Use C (x) to approximate C when x = 1 and x = 3. Is this a good approximation? (ii) Use C (x) to approximate C when x = 1 and x = 50. Is this a good approximation? (b) (Example 5 Chapter 6.6) In a precision manufacturing process, ball bearings must be made with a radius of 0.6mm, with a maximum error in the radius of ±0.015mm. Estimate the maximum error in the volume of the ball bearing. Hint: The volume of a sphere is given by V = 4 3 πr3 where r is the radius. (c) (Exercise 34 Chapter 6.6) A cubical crystal is growing in size. Find the approximate change in the length of a side when the volume increases from 27 cubic mm to 27.1 cubic mm. (d) (Exercise 29 Chapter 6.6) A sphere has a radius of 5.81 inches with a possible error of ±0.003 inches. Estimate the maximum error in the volume of the sphere. Solution 9.1. (a) (i) C (x) = 6x. Linear approximation (in the form useful for us here) is C C (x) x. Hence we have C C (3) x = 18 1 = 18. This is not a particularly good approximation, the change in x ( x) should be small, and in particular small compared to x itself. Here, the change in x is quite large, of the same order of magnitude of x. Repeat this process for part (ii). (b) Here we use the form for linear approximation f = f(x 0 + δx) f(x 0 ) f (x 0 )(x x 0 ), where x 0 is fixed (the point we know). In this case, the function is the volume V (r) = 4 3 πr3, in the variable r (the radius). We want to determine the possible variation in V with the variation in r. Here r 0 = 0.6 and δr = We also need V (r) = 4πr 2. Therefore V V (0.6) = 4π mm 3. 1

2 2 REVIEW FOR PRELIM 1 SOLUTIONS (c) Here we re given the change in volume and want to know the change in side length. This means we need to write the side length l as a function of the volume V. We know that for a cube, V (l) = l 3. Therefore l(v ) = 3 V, and l (V ) = 1V 2 3. The change 3 in V is 0.1, and our We use the linear approximation from part (b) to get l l (27) 0.1 = = = mm. (d) Follow the process of part (b). Tutorial 9.2. Derivatives of Exponentials (a) (Exercise 52 chapter 4.4) The age/ weight relationship of female Arctic foxes caught in Svalbard, Norway, can be estimated by the funtion M(t) = 3102e e 0.022(t 56) where t is the age of the fox in days and M(t) is the weight of the fox in grams. (i) Estimate the weight of a female fox that is 200 days old. (ii) Use M(t) to estimate the largest size that a female fox can attain (Hint: Find lim t M(t)). (iii) Estimate the age of a female fox when it has reached 80% of its maximum weight. (iv) Estimate the rate of change in weight of an Arctic fox that is 200 days old. (b) Differentiate the following functions: (i)f(t) = 2 3 t (ii) f(t) = et e t (iii) f(t) = tet +2 t e 2t +1 (c) The value of a particular investment changes over time according to the function S(t) = 5000e 0.1(e0.25t), where S(t) is the value after t years. Which of the following is the rate at which the value of the investment is changing after 8 years? (i) 618 (ii) 1934 (iii) 2011 (iv) 7735 (v) (d) It has been observed that there has been an increase in the proportion of medical research papers that use the word novel in the title or abstract, and that this proportion can be accurately modeled by the function p(x) = e x where x is the number of years since (i) Find p(40).

3 REVIEW FOR PRELIM 1 SOLUTIONS 3 (ii) If this phenomenon continues, estimate the year in which every medical article will contain the word novel in its title or abstract. (iii) Estimate the rate of increase in the proportion of medical papers using this word in the year Solution 9.2. (a) Don t worry about the numbers so much here, make sure you can understand what you re expected to compute. (i) Compute M(200), ie sub in 200 into the formula for M(t). (ii) The first thing to remember is the shape of the function e t. We have lim t e t = 0 We have hence lim 0.022(t 56) = t lim M(t) = t 3102e0 = Therefore the largest size that a fox can attain is kg by this model. (iii) We want to find the (approximate) value of t such that M(t) = 80/100M max where M max = 3102 by the previous question. Therefore we need to solve 3102e e 0.022(t 56) = e e 0.022(t 56) = 0.8 e 0.022(t 56) = ln (t 56) = ln( ln 0.8) t = 56 ln( ln 0.8) (iv) Compute M (t) = e 0.022(t 56) e e 0.022(t 56) - use the chain rule twice. Then simply compute M (200). (b) Recall the chain rule, the product rule and the quotient rule for the following: (i) f (t) = 2 (e t ln 3 ) = 2 ( t ln 3) e t ln 3 = 2 ln 3 2 t 3 t = ln 3 t 3 t (ii) f (t) = (et + e t )t (e t e t ) = et (t 1) + e t (t + 1) t 2 t 2

4 4 REVIEW FOR PRELIM 1 SOLUTIONS (iii) f (t) = (et + te t )(e 2t + 1) (te t + 2)(2e 2t + 1) (e 2t + 1) 2 (you could simplify this somewhat). (c) Compute S (t) using the chain rule twice. You should obtain S (t) = 125e 0.25t e 0.1e0.25t It remains to compute S (8) , so the answer is (ii). (d) Again, try not to be put off by the numbers given, do you know what to compute in each question? (i) Use a calculator to compute p(40). (ii) The year when every paper contains the word novel would be a year when the proportion is equal to 1. Therefore we need to solve 1 = e x e x 1 = x = ln = ln x = 1 ln years Tutorial 9.3. Derivatives of logarithmic functions (a) Differentiate the following functions: (i) f(x) = ln x 3 (ii) f(x) = 10 x log 10 x (iii) f(t) = e t + ln 2t (b) (Exercise 65 chapter 4.5) Suppose that the population of a certain collection of rare Brazilian ants is given by P (t) = (t + 100) ln(t + 2) where t represents the time in days. Find the rates of change of the population on the second day and on the eigth day. (c) (Exercise 67 chapter 4.5) The following formula shows a relationship between the amount of energy released and the Richter number. M = 2 3 log E where E is measured in kilowatt-hours. (i) For the 1933 earthquake in Japan, what value of E gives a Richter number M = 8.9?

5 REVIEW FOR PRELIM 1 SOLUTIONS 5 (ii) If the average household uses 247 kwh per month, how many months would the energy released by an earthquake of this magnitude power 10 million households? (iii) Find the rate of change of the Richter number M with respect to energy when E = 70, 000 kwh. (iv) What happens to dm as E increases? de Solution 9.3. (a) You will generally need the chain rule for these questions. { x 3 if x 3 (i) Note that x 3 =. This means the 3 x if x < 3 solution could depend on whether or not x 3. You should encounter this as you take derivatives, though the effects cancel and there is a single solution: f (x) = 1 2(x 3) ln x 3 (ii) Remember 10 x is NOT of the form x n, you CANNOT use the power rule here. You will need to use the product rule. (iii) f (x) = ln x log 10 x + f (t) = 0.5 ( e t + 1 t ) e t + ln 2t 10x x ln 10 (b) P (t) = ln(t + 2) + t+100 t+2. Compute P (2) and P (8). (c) To get the final answers here you can use a calculator. (i) We want to solve Rearranging, we get 8.9 = 2 3 log E E = (ii) The number of months is given by E/( ), where E is the energy found in the previous question. (iii) First, we compute the derivative M (E) = 2 1. Evaluate 3 ln 10 E this for E = (iv) The rate of change of M is proportional to 1/E, hence it decreases towards zero as E increases.

6 6 REVIEW FOR PRELIM 1 SOLUTIONS Tutorial 9.4. (Exercise 55 Chapter 5.2) The mathematical relationsip between the age of a captive female moose and its mass can be described by the function M(t) = 369(0.93) t t 0.36, t 12 where M(t) is the mass of the moose (in kilograms) and t is the age (in years) of the moose. Find the age at which the mass of a female moose is maximised. What is the maximum mass? Solution 9.4. We need to compute the critical points of M. These are given by the first derivative: M (t) = 369[ln t t t 0.36t ] = t t 0.36 [ln t 1 ] The point of factorising this way is to examine the factors individually to see when they re zero (we have no problems of M (t) not being defined at a point). Therefore the critical points are t 0 = 0 and t 1 = Compute the sign of the derivative on either side of the ln 0.93 second critical point. You should find that it is positive before and negative after t 1, hence by the first derivative test, this is a relative maximum of M. Compute M(t 1 ) to find this maximum mass. Tutorial 9.5. Consider the function f(x) = 0.008x x x + 7. Find the domain of f, the critical points of f, the intervals on which f is increasing and relative minima and maxima (use the second derivative test then the first derivative test). Solution 9.5. Recall: domain is where the function makes sense, critical points are where the derivative is not defined or is zero. f is increasing on intervals where the derivative is positive, decreasing where the derivative is negative. The sign of the second derivative evaluated at the critical points gives the relative maxima or minima. Tutorial 9.6. (Exercise 58 Chapter 5.2) Researchers determined that the ratings people gave for a film could be approximated by R(t) = 20t t where t is the length of the film (in minutes). find the film length that received the highest rating. Solution 9.6. Proceed in the same way as in question 9.4.