f x (prime notation) d dx

Transcription

1 Hartfield MATH 040 Unit Page Basic Techniques for Finding Derivatives In the previous unit we introduced the mathematical concept of the derivative: f lim h0 f( h) f ( ) h (assuming the limit eists) In this unit we will look at rules for finding derivatives that will be simpler than applying the definition. There are many different forms of notation used to indicate a derivative. For eample, if we want the derivative of the function f(), we could epress it as: f (prime notation) d f d (Leibniz notation) D f (subscript notation) Assuming that y = f(), a common variation on Leibniz notation is dy d.

2 Hartfield MATH 040 Unit Page Eample: Epress all appropriate forms of notation that could be used for the derivative of the function g(t). The notation d may sometimes be used to d indicate that you wish to find a derivative without defining an epression eplicitly as a function.

3 Hartfield MATH 040 Unit Page 3 Rules For Differentiation Rule 1: Rule : Constant Rule For any real constant k, d k d 0. (If f = k, then Power Rule f = 0.) For any constant eponent n, d d n (If f = n1 n. n, then f = n1 n.) Eample set 1: Find each derivative. d A-1. 4 d A-. f 13 B-1. d 4 d B-. f 7

4 Hartfield MATH 040 Unit Page 4 Eample set : Find each derivative. Eample set 3: Find each derivative. C-1. d d 3 d 1 D-1. d C-. f 4 5 D f

5 Hartfield MATH 040 Unit Page 5 Practice: Find the derivative of each function. A. f 1 B. f 1 C. f 3 Rule 3: Rule 4: Constant Multiple Rule For any constant k, d k f k f d f = kg, k g.) (If then f = Sum/Difference Rules d f g f g d (If f = u v, then f = u v.).

6 Hartfield MATH 040 Unit Page 6 Eample set 1: Find each derivative. d d A Two noteworthy shortcuts based on rules and 3: d d 1 and d c c d A-. f 6 8 Eample set : Find the derivative. d d B-1. 3 d d 7 C-1. B-. f 6 5 C-. 4 f 85

7 Hartfield MATH 040 Unit Page 7 Eample: Find f. Eample: Find f. f f 4 1 6

8 Hartfield MATH 040 Unit Page 8 Practice: Find f. Practice: Find f. f f 5 3

9 Hartfield MATH 040 Unit Page 9 Marginal Analysis We will use the following function notations for application problems in business and economics: Revenue Function R() = Total revenue from selling units Cost Function C() = Total cost of producing units Profit Function P() = Total profit from producing and selling units Economists use the term marginal to refer to rates of change. The derivative, which coincides with instantaneous rate of change, is used when talking about marginal in calculus. If you have a cost function at some level of production, the marginal cost is the epected additional cost of producing the ( + 1)st unit. That is, if you are already making units, the marginal cost predicts the cost of making the net unit. Notationally, when C() represents the cost function, C () represents the marginal cost function.

10 Hartfield MATH 040 Unit Page 10 It is important to understand that the marginal function may not eactly identify the additional cost of the ( + 1)st unit. The actual cost of that net unit can be eactly found by calculating C( + 1) C(). In most cases though, the evaluation of the marginal cost function at is very close to the eact value found by the difference. Analogous statements can be made for revenue and profit. Some additional notes for future reference: Analogous statements can be made for revenue and profit. Thus R () represents the marginal revenue function and P () represents the marginal profit function. A common function in economics is the demand function p Dq which relates the number of units q that consumers are willing to purchase at price p. The revenue generated from selling q units is then found R q q D q. by

11 Hartfield MATH 040 Unit Page 11 Eample: A steel mill determines that its cost 3 function is C( ) dollars,where is in the daily production of tons of steel. A. Find the cost of manufacturing 64 tons of steel per day. B. Find the marginal cost function. C. Find the marginal cost of producing one more ton when 64 tons are being produced. D. Calculate the actually cost of producing one more ton by finding the cost of manufacturing 65 tons.

12 Hartfield MATH 040 Unit Page 1 Eample: If the demand function for 500 q heavyweight paper is p 5 dollars, where q is in reams, answer the following: C. Find the marginal revenue function. A. Find the revenue function. B. Find the revenue generated from 00 reams being sold. D. Calculate and interpret the marginal revenue function when 00 reams are being sold.

13 Hartfield MATH 040 Unit Page 13 Practice: Continuing the previous eample, suppose the cost function in dollars for heavyweight paper is given by C( q) 10 4 q, 0 q 300. C. Calculate and interpret the marginal profit function when 100 reams, 00 reams, and 50 reals are being produced and sold. A. Find the profit function. B. Find the marginal profit function.

14 Hartfield MATH 040 Unit Page Derivatives of Products and Quotients Rule 5: Product Rule Eample: Find f. d f g d f g g f f( ) 4 (If f = u v, then f = u v v u.)

15 Hartfield MATH 040 Unit Page 15 Practice: Find f. Rule 6: Quotient Rule f( ) 5 3 d f g f f g d g g (If f = then u f = v and v 0, vu uv v.)

16 Hartfield MATH 040 Unit Page 16 Eample: Find f. Practice: Find f. f ( ) 1 f ( ) 1 1

17 Hartfield MATH 040 Unit Page 17 Average Cost and Marginal Average Cost For a cost function C where C represents the cost of manufacturing items, the average cost function C is found by C / and determines the average cost per item. It is possible to find a marginal with respect to an average function. As with other marginals, a marginal average function is predictive of the change occurring when you increase the number of items by one. The marginal average cost function C is the derivative of the average cost function and finds the rate of change in the average cost. Analogous statements can be made for revenue and profit. Thus R represents the average revenue function, determined by R representing the marginal R /, with P represents average revenue function. the average profit function, determined by P /, with P representing the marginal average revenue function.

18 Hartfield MATH 040 Unit Page 18 Eample: The total profit (in tens of dollars) from selling self-help books is 5 6 P ( ). 3 C. Evaluate P, P, and P when = 0. Interpret each evaluation. A. Find the average profit function. B. Find the marginal average profit function.

19 Hartfield MATH 040 Unit Page 19 Practice: The fuel economy (in m.p.g.) of a Porsche driven at a speed of m.p.h. 000 is E ( ). 305 B. Evaluate E and E when = 80, rounding logically. Interpret each evaluation. A. Find E ( )

20 Hartfield MATH 040 Unit Page The Chain Rule Recall from algebra that composed functions consist of one function inside a second function. For eample, 3 1 is considered to be a composite function because the function ² 1 eists within a cubing function (that is, the output of ² 1 is the input to the cube). We can decompose 3 1 by defining the two functions that are brought together to make the new function. 3 1 f g( ) with 3 f ( ) g( ) 1 We tend to call g the inside function and f the outside function. Eample: Decompose each of the following functions so that f(g()) returns the original function. 3 6

21 Hartfield MATH 040 Unit Page 1 Practice: Decompose each of the following functions so that f(g()) returns the original function. A. 1 4 Finding a derivative for a function created by a composition requires we consider the differentiation of both the inside and outside function. It turns out that the outside derivative is taken initially without regard to the inside epression, with the inside derivative being multiplied in separately. This is called the Chain rule. B. 7

22 Hartfield MATH 040 Unit Page Chain Rule Rule 7: Chain Rule Eample: Find y. d f g ( ) f g ( ) g ( ) d A. 3 6 y We can also restate the composition y f g( ) as y = f(u) and u = g(). Their respective derivatives would be dy du f() u and g ( ). du d By appropriate substitution, we can present Chain Rule using Leibniz s Notation: B. y 7 dy d dy du. du d

23 Hartfield MATH 040 Unit Page 3 Practice: Find y. y 1 A. 4 Frequently you may need to combine Chain Rule with Product Rule or Quotient Rule to find a derivative. Eample: Find y. B. y y

24 Hartfield MATH 040 Unit Page 4 Eample: Find y. Practice: Find y. y 1 y

25 Hartfield MATH 040 Unit Page 5 Applications Eample: Suppose that for a group of 10,000 people, the number who survive to age is N( ) Evaluate and interpret N and N when = 36.

26 Hartfield MATH 040 Unit Page 6 Practice: A 35 year old male of average weight is injected with a 100 cubic centimeters of a specific medication. At t hours after injection, the body is 00t metabolizing Vt () cc of the t 1 3 medication. Evaluate and interpret V and V at t = 4.

27 Hartfield MATH 040 Unit Page Derivatives of Eponential Functions Rule 8: Eponential Rule (if base is e) d e e d Rule 8*: Eponential Rule (if base is a) d a ln a a d Our primary but not eclusive focus will be on differentiating eponential epressions with a base of e. Frequently you will need to find derivatives where the eponent of a base e (or a) eponential is not simply. Strictly speaking this creates a composition and requires the use of Chain Rule. We can integrate Chain Rule with the Eponential Rule as follows: Rule 8a: d e g e g g d d a g g ln a a g d

28 Hartfield MATH 040 Unit Page 8 Eample: Find the derivative of each. Practice: Find the derivative of each. A. 4 1 y 10e A. y 6 5e B. y 3 e 4 3 B. y 6 3 8e C. 3 y 5 C. 1 y 43

29 Hartfield MATH 040 Unit Page 9 Eample: Find the derivative. Eample: Find the derivative. y e y e

30 Hartfield MATH 040 Unit Page 30 Practice: Find the derivative. Practice: Find the derivative. y e e y e 1

31 Hartfield MATH 040 Unit Page 31 Applications: Eample: A cup of coffee brewed at 00 degrees, if left in a 70-degree room, will cool to T(t) = e 0.04t ( F) in t minutes. Determine the temperature of the coffee in 1 hour and the rate of change in the temperature at that time.

32 Hartfield MATH 040 Unit Page 3 Eample: For a particular market the demand 0.1q function of an item is p 00 e, where q is in thousands of units. Find the revenue function and its derivative. Then evaluate both and interpret when 5 thousand units are being sold.

33 Hartfield MATH 040 Unit Page Derivatives of Logarithmic Functions Rule 9: Logarithmic Rule (if base is e, > 0) d d ln 1 Rule 9*: Logarithmic Rule (if base is a, > 0) d d log a 1 lna Similar to eponentials, our primary but not eclusive focus will be on differentiating logarithmic epressions with a base of e. Recall that the domain of a logarithmic function is based on when the argument of the log is positive. As with eponentials, frequently you will need to find derivatives where the argument is not simply. Again this creates a composition and requires the use of Chain Rule. We can integrate Chain Rule with the Logarithmic Rule as follows: Rule 9a: d g ln g, d where g > 0 g d g log a g d ln( a) g

34 Hartfield MATH 040 Unit Page 34 Eample: Find the derivative of each. yln 8 3 A. Practice: Find the derivative of each. A. 3 yln 1 B. 3 y ln 5 B. y log y log 4 C.

35 Hartfield MATH 040 Unit Page 35 Eample: Find the derivative. Eample: Find the derivative. y 3 ln y 3 ln

36 Hartfield MATH 040 Unit Page 36 Practice: Find the derivative. Practice: Find the derivative. y e ln y ln 1

37 Hartfield MATH 040 Unit Page 37 Applications: Eample: The total revenue (in thousands of dollars) produced by selling thousands of books can be epressed as R( ) 50ln4 1. The cost (in thousands of dollars) to produce thousands of book is given by C( ) 5. B. Find the profit function and the marginal profit function. Interpret both when 10 thousand books are being sold. A. Find the marginal revenue function and interpret it when 10 thousand books are being sold.

38 Hartfield MATH 040 Unit Page 38 Eample: Based on projections from the Kelly Blue Book, the average resale value of a 010 Toyota Corolla sedan can be anticipated by the function f( t) logt 1, where t is the number of years since 010. Find & interpret f and f when t = 4.