MATH 151, Fall 2013, Week 10-2, Section 4.5, 4.6

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1 MATH 151, Fall 2013, Week 10-2, Section 4.5, 4.6 Recall the derivative of logarithmic and exponential functions. Theorem 1 (ln x) = (ln f(x)) = (log a x) = (log a f(x)) = Theorem 2 (a x ) = (a f(x) ) = Section 4.5 Exponential Growth/Decay In many natural phenomena, quantities grow or decay at a rate proportional to the size. For instance, if f(t) is the number of population of a certain animals or bacteria at time t, it seems reasonable to expect that the rate of growth f (t) is proportional to the population f(t) under ideal conditions; that is f (t) = k f(t), for some constant k Actually the above model provides fairly accurate prediction in many areas such as nuclear physics, chemistry, finance, etc. Theorem If a function f(t) satisfies then it is of the form f(t) = A e kt, for some constants A, k Note that the initial quantity is determined by f(0) = A e 0 =. Useful fact: e ln a =. For example, e (ln 5.4)t =. 1

2 Ex 1. A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 240. (a) Find an expression for the number of bacteria after t hours. (b) Find the number of bacteria after 4 hours. (Round it to the nearest whole number.) (c) Find the rate of growth after 4 hours. (Round it to the nearest whole number.) (d) When will the population reach 10, 000? (Round it to one decimal place.) Ex 2. (Half-life) A sample of a radioactive substance decayed to 90.5% of its original amount after a year. (Round your answers to two decimal places.) (a) What is the half-life of the substance? (b) How long would it take the sample to decay to 70% of its original amount? 2

3 Ex 3. Bismuth-210 has a half-life of 5.0 days. A sample originally has a mass of 800mg. (a) Find a formula for the mass remaining after t days. (b) Find the mass remaining after 40 days. (c) When is the mass reduced to 1 mg? (d) Sketch the graph of the mass function. (Do this on paper. Your teacher may ask you to turn in this work.) 3

4 Newton s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and the surroundings. (we assume the temperature of surroundings does not change) Ex 4. Suppose you are having a cup of coffee which takes half an hour to cool from 190 F to 110 F in a room that is kept at 70 F. (a) What was the temperature of the coffee after 15 minutes? (b) How long will it take the coffee to cool down 80 F. 4

5 Ex 5. A roast turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a room where the temperature is 75 F. (a) If the temperature of the turkey is 150 F after half an hour, what is the temperature after 50 minutes? (b) When will the turkey have cooled to 105 F? Ex 6. How long will it take an investment to double in value if the interest rate is 5% compounded continuously? 5

6 Ex 7. The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At a specific temperature the pressure is 101.3kP a at sea level and 87.1kP a at h = 1, 000m. (a) What is the pressure at an altitude of 3000m? (b) What is the pressure at the top of a mountain that is 8848m high? Ex 8. (19, Exam III of Spring 2013) A metal ball cools at a rate equal to one-third the difference between its temperature and the temperature of its surroundings. Suppose the ball is taken from outdoors where the temperature is 34 C to a room where the temperature is 19 C. (a) (5 pts) Find the temperature of the ball after 2 minutes. (b) (5 pts) When will the ball have a temperature of 25 C? 6

7 Section 4.6 Inverse Trigonometric Functions Sketch graphs of the following functions y = sin x y = cos x y = tan x. They are not one-to-one, find restricted domains to make them one-to-one. For those restricted domain, find the corresponding ranges. Dom(sin x) = Ran(sin x) = Dom(cos x) = Ran(cos x) = Dom(tan x) = Ran(tan x) = Definition Inverse trigonometric functions are defined by sin 1 x = y sin y = x, π 2 y π 2 cos 1 x = y cos y = x, 0 y π tan 1 x = y tan y = x, π 2 < y < π 2 Notation: arcsin = sin 1, arccos = cos 1, arctan = tan 1 7

8 Ex 9. Find the exact values of each expression.(enter your answer in radians.) (a) sin 1 ( ) (b) cos 1 ( 2 ) (c) tan 1 ( 3) (d) cos 1 ( 1 2 ) Ex 10. (a) sin(arcsin(.6)) (b) tan(arctan(10)) (c)arcsin(sin( π 12 )) (d) sin 1 (sin( 7π 3 )) (e) arccos(cos(4π 3 )) (f) arcsin(sin(2π 3 )) Identities in inverse trig. functions sin(sin 1 x) = x, x sin 1 (sin x) = x, only when x [ π 2, π 2 ] cos(cos 1 x) = x, x cos 1 (cos x) = x, only when x [0, π] tan(tan 1 x) = x, x tan 1 (tan x) = x, only when x ( π 2, π 2 ) 8

9 Constructing a triangle Suppose sin 1 x = y. Consider a right triangle below. sin 1 x = y sin y = x, so y is an angle. Set A = y. Convention: hypotenuse = 1 sin y = x height = x. A Use Pythagorean thm to label the base. Now one can calculate cos y and tan y. cos y = tan y = Ex 11. Simplify the expression tan(sin 1 x). Ex Simplify the expression tan(cos 1 x). A 9

10 Derivative of Inverse Trig. Functions To find d dx (sin 1 x), set sin 1 x = y. So we have sin y = x Apply implicit differentiation to the above to obtain cos y dy dx = 1 or dy dx = 1 cos y. One can construct a triangle for sin 1 x = y, to show cos y = (sin 1 x) = 1 cos y =.. (Ex 11.) So For d dx (cos 1 x) we can start with cos 1 x = y (cos y = x) to have From Ex 11-1, we know sin y = sin y dy dx = 1 or dy dx = 1 sin y (cos 1 x) =. So 1 sin y =. Similarly we also have tan 1 x = y tan y = x dy dx = 1 sec 2 y = cos2 y =. Theorem (sin 1 x) = (cos 1 x) = (tan 1 x) = 10

11 Ex 12. Find the derivative of the following functions (a) y = arctan( x) (b) y = arccos(e 9x ) (c) y = sin 1 (2x + 1) (d) y = (tan 1 (9x)) 2 (e) f(x) = arctan( 3 x ) (f) g(x) = arcsin(ln x) Ex 13. Simplify the following functions (Problems from past common exams). (a) tan(arccos 1 4 ) (b) cot(arcsin(3 4 )) (c) sin(tan 1 ( x 2 )) 11