UNIT 5: DERIVATIVES OF EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS 5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS, y = e X Qu: What do you remember about exponential and logarithmic functions? e, called Euler s number or the natural number, is a special irrational number, like π, that has many applications in the world of mathematics. e, which we round to approximately 2.718, has particular relevance to exponential functions found in nature and in economics, and has special properties in the realm of calculus. e = lim 1+ f(x) = e x operates similarly to other exponential functions; like other exponential functions, its inverse is a logarithmic function: y = log. Because e is called the natural number, its inverse is called the natural logarithm function, and is given a special designation: y = ln x. 8 6 f( x) = e x g( x) = ln( x) h( x) = x 4 2-15 -10-5 5 10 15-2 -4-6 For f(x) = e x, For f(x) = ln x, D: {xєr} D: {x x > 0, xє R} R: {y y > 0, yєr} R: {yєr} Y- intercept is (0, 1) x-intercept is (1, 0) e ln x = x, x > 0 ln e x = x, xєr. horizontal asymptote at y = 0. Vertical asymptote at x = 0.
Derivatives: (keep in mind that all previously learned rules for differentiation apply) The derivative of f(x) = e x is f (x) = e x. This function is special because it is its own derivative. If we have a composite function, f(x) = e g(x), the derivative is f (x) = e g(x) g (x). (Chain Rule) Ex. Determine f (x) for the following. Evaluate each at -2. Leave e as an exact value. a) g(x) = b) f(x) = x 2 e x c) h(x) = 3 d) i(x) = e) j(x) = Ex. p. 233 #8
5.2 THE DERIVATIVE OF THE GENERAL EXPONENTIAL FUNCTION, y = b x Recall that exponential functions are used to model situations in which there is rapid growth or rapid decay: business and financial models, bacterial growth, the spread of disease, earthquakes, and nuclear decomposition are all examples of exponential relations. Clearly, it is to our benefit to be able to discuss the instantaneous rate of change (derivative) of such models. For f(x) = b x f (x) = b x (ln b) (note: There is a good proof for this in your text it s just not necessary for our purposes) For composite functions, f(x) = b g(x), f (x) = b g(x) (ln b)g (x). Ex. Differentiate each of the following: a) y = 2 3x b) s = 10 3t-5 c) y = 400(2) x+3 Ex. A certain radioactive material decays exponentially. The percent, P, of the material left after t years is given by the formula P(t) = 100 (1.2) -t. a) Determine the half-life of the substance. b) How fast is the substance decaying at the point where the half-life is reached?
UNIT 5 SUPPLEMENT: REVIEW OF LOGS AND EXPONENTS,AND CALCULUS INVOLVING LOGARITHMS REVIEW: Exponent Laws = ; = ; = ; = ; = = ; = ; = ; Logarithm Laws: Radian Measure: = 1) log = 2) = 3) log (AB) = log A + log B 4) log (A/B) = log A log B 5) To solve an exponent, use logs: 1.4 = 2 Log 1.4 = log 2 Log 1.4 = x log 2 1.4 2 = BASIC EXPONENTIAL EQUATION: =, where c = initial amount; a = growth or decay factor (expressed as a decimal); x = # growth or decay periods; y = final amount. Half-Life: the amount of time necessary for a substance to decay by half. *For optimization problems, follow the same steps as in Unit 3. Be sure to identify the quantity to be optimized, determine the appropriate formula, and use the derivative to find the minimum/maximum value.
5.2 APPENDIX 1) Natural Log Functions The Natural number, e, is used in a particular exponential function: =. The inverse of this is the Natural log function (ln): = log =. We write this as y = ln x. To take the derivative of y = ln x: - Use the properties of inverse functions: = - Replace ln x: = - Use Implicit Differentiation to derive both sides: = 1 = = 1 Therefore, For the natural log (ln) function y = ln x, =, >0 For y = ln g(x), = 2) General Log Functions If f(x) = log, = If f(x) = log, = 3) Logarithmic Differentiation We have thus far seen two different types of differentiation in Calculus: explicit, where y is defined explicitly in terms of x (y = f(x)), and implicit, where y is not defined precisely (ex. = +25). We can consider another type of differentiation, where we use the properties of logs to find the derivative. Ex. Derive =
Sol. Take the ln of both sides. ln y = x ln x =ln + x ln x. use properties of logs to drop the exponent down use implicit differentiation for y; use product rule for = +1 = +1 replace y with its original definition of. Ex. Use the rules above to derive the following. a) =ln 5 b) = c) =ln + d) y = log e) =5log 2 +3 f) = +3 Ex. Determine the equation of the line tangent to = at the point (1, 0). Ex. Find the minimum of f(x) =. Ex. Derive = at x = -1. (NOTE: although it is not required this semester to learn logarithmic differentiation, it is helpful to know in order to solve this type of derivative.)
5.3 OPTIMIZATION PROBLEMS INVOLVING EXPONENTIAL FUNCTIONS (Note: reviewing Chapter 3 will refresh you on optimization). Recall that when solving a question involving optimization: 1) You are looking to minimize or maximize a function by determining the appropriate values of the independent variable. 2) When given a problem, understand what it is asking you identify quantities that can vary. Determine a function in one variable that represents the quantity to be optimized. 3) Determine the domain of the function to be optimized, using the information given in the problem. 4) Use the algorithm for finding extreme values to find the absolute maximum or minimum value of the function on the domain. 5) Use your results to answer the original problem. Check for logic and sensibility of your answer.
5.4 THE DERIVATIVES OF SINE AND COSINE ( y = sin x and y = cos x) t 1 = 1.00 f( x) = sin( x) 4 3 2 1-8 -6-4 -2 2 4 6 8-1 -2-3 -4 f( x) = cos( x) 3 2 1-6 -4-2 2 4 6-1 -2-3 From studying the slopes of the tangent lines over sine and cosine, we can understand the derivatives of each. f(x) = sin x f (x) = cos x f(x) = cos x f (x) = -sin x For composite functions, we use the Chain Rule: If y = sin f(x), y = cox f(x) f (x). If y = cos f(x), y = -sin f(x) f (x).
Ex. Differentiate the following functions: a) y = sin x 2 b) y = x cos x c) y = sin 2 x d) y = cos 3x e) y = cos (1 + x 3 ) sin x + cos x f) y = e Ex. Determine the equation of the tangent line to the graph of y = x cos 2x at the point x =. Ex. Determine the maximum and minimum values of the function f(x) = cos 2 x on the interval x Є [0, 2π].
5.5 THE DERIVATIVE OF y = TAN X Using the knowledge that tan x = sin x / cos x, let us use derivative rules to find the derivative of tan x. The derivative of y = tan x is therefore sec 2 x. For composite functions of the type y = tan f(x), y = sec 2 f(x) f (x). Ex. Differentiate the following. a) y = tan (x 2 + 3x) b) y = (sin x + tan x) 4 c) y = x tan (2x 1)