Calculus Overview. f(x) f (x) is slope. I. Single Variable. A. First Order Derivative : Concept : measures slope of curve at a point.

Transcription

1 Calculus Overview I. Single Variable A. First Order Derivative : Concept : measures slope of curve at a point. Notation : Let y = f (x). First derivative denoted f ʹ (x), df dx, dy dx, f, etc. Example : f (x) = 2 + 3x 4x 2 f ʹ (x) = 3 8x. f(x) f (x) is slope Calculus Overview Page 1 x

2 Rules of Differentiation : Constant Rule : f (x) = a f ʹ (x) = 0 Sum Rule : f (x) = g(x) + h(x) f ʹ (x) = g'(x) + h'(x) Product Rule : f (x) = g(x)h(x) f ʹ (x) = g ʹ (x)h(x) + h ʹ (x)g(x) Quotient Rule : f (x) = g(x) h(x) f ʹ (x) = Polynomial Rule : f (x) = ax n f ʹ (x) = anx n 1 g ʹ (x)h(x) h ʹ (x)g(x) h(x) 2 Chain Rule : f (x) = g(h(x)) f ʹ (x) = g ʹ (h(x)) h ʹ (x) Exponential / Log Function Facts : lna x = x ln a lna = b e b = a Exponential Function Rule : f (x) = e x f ʹ (x) = e x e x e y = e x +y Natural Log Rule : f (x) = ln x f ʹ (x) = 1 x ln(ab) = ln a + lnb e ln x = x Calculus Overview Page 2

3 B. Second Order Derivative Concept : measures curvature (how fast slope is changing). Notation : Let y = f (x). Second derivative denoted f ʹ (x), d 2 f dx, d 2 y 2 dx, f, etc. 2 Cases : Concave f ʹ (x) < 0 Convex f ʹ (x) > 0 Point of inflection f ʹ (x) = 0 Example : f (x) = 2 + 3x 4x 2 f ʹ (x) = 3 8x f ʹ (x) = 8 concave for all x Calculus Overview Page 3

4 C. Optimization(Maximum, Minimum) f(x) max First Order (Necessary) Condition : f ʹ (x) = 0. If f ʹ (x) > 0, can increase f (x) by increasing x. If f ʹ (x) < 0, can increase f (x) by decreasing x. concave x Second Order (Sufficient) Condition : f ʹ (x) < 0, sufficient for maximum. f ʹ (x) > 0, sufficient for minimum. f(x) convex Example : f (x) = 2 + 3x 4 x 2 f ʹ (x) = 3 8x f ʹ (x) = 0 3 8x = 0 x * = 3/8 f ʹ (x) = 8 < 0 f (x) concave x * is a maximum. min x Calculus Overview Page 4

5 II. Multiple Variables A. Multivariable Functions Notation : Function with more than one independent variable, y = f (x 1, x 3,x 4,..., x n ). In this course, we will mostly work on functions with two independent variables, y = f (x 1 ) x 2 y x 1 x 2 x 1 Calculus Overview Page 5

6 B. Partial Derivatives Idea : While taking the partial derivative with respect to x i, we have to treat other variables constant. The intuition is we keep the values of all other variables fixed. Notation : y = f (x 1,x 3,x 4,...,x n ) f = f x1 = f 1 f = f x2 = f 2 : : f x n = f xn = f n Calculus Overview Page 6

7 Examples : 1) f (x 1 ) = 2x 1 + 3x 2 x 1 2 x 2 f = f = 2) C(L,K) = 4L 2 + 3K 2, C is a cost function depending on two inputs : Labor amount (L) and Capital amount (K) C L = C K = Calculus Overview Page 7

8 C. Higher Order Partials (functions with two - variables) f = 2 f x = second partial 2 wrt* x 1 = f x1 x 1 = f 11 1 f = 2 f x = second partial wrt x 2 2 = f x2 x 2 = f 22 2 f = 2 f = cross partial wrt x 1 and x 2 = f x2 x 1 = f 12 f = 2 f = cross partial wrt x 2 and x 1 = f x1 x 2 = f 21 * = with respect to (wrt) Calculus Overview Page 8

9 Example : f (x 1,x 2 ) = 2x 1 + 3x 2 x 1 2 x 2 f 1 = f 11 = f 2 = f 22 = f 21 = f 12 = Note : This example illustrates the mathematical result that the order in which partial differentiation is conducted to evaluate second - order derivatives does NOT matter. Calculus Overview Page 9

10 D. Total Differentiation If all the x's are varied by a small amount, the total effect on y will be the sum of effects. In other words total change in y can be decomposed into changes resulting from changes in x 1, x 3,..., x n. definition : Let y = f (x 1,x 2,x 3,...,x n ). A total total differentiation of f is defined as : df = f dx 1 + f dx f x n dx n meaning of f x i dx i ( for any i {1,2,3,...,n}) : how much f changes when x i. changes by dx i. Calculus Overview Page 10

11 Let's play with total differentiation equation for functions with two variables (i.e y = f (x 1,x 2 )). df = f dx 1 + f dx 2, divide both sides by dx 1. df = f + f dx 2 dx 1 dx 1 The first term ( f ) shows the direct effect of x 1 on f, and the second term ( f dx 2 dx 1 ) shows the indirect effect of x 1, via the impact on x 2. Calculus Overview Page 11

12 Example : If F(K,L) = 25KL and K = L 2, what is the total effect of a change in L on F(L,K)? Calculus Overview Page 12

13 E. Unconstrained Optimization : idea : discuss the conditions for optimal values of x i 's that maximizes/minimizes y = f (x 1,x 2,x 3, x 4,..., x n ) First Order (Necessary) Condition : For all i {1,2,3,...,n}, f x i = 0 Intuitively, if one of the partials were greater or less than 0, then y could be increased by increasing or decreasing x i. NOTE : Second order condition will be complex that would be beyond the needs of this course. Calculus Overview Page 13

14 F. Constrainted Optimization : Suppose you want to find the values x 1, x 3, x 4,..., x n maximizes/minimizes subject to y = f (x 1, x 3, x 4,..., x n ) g(x 1,x 3, x 4,...,x n ) = c. First, we set up Lagrangian L = f (x 1, x 3, x 4,..., x n ) + λ[g(x 1, x 3, x 4,..., x n ) c], where λ is the additional variable called the Lagrange multiplier (λ will be treated as a variable in addition to x's). The conditions for a optimal point are : L = f 1 + λg 1 = 0 : : L = f 2 + λg 2 = 0 L x n = f n + λg n = 0 L λ = g(x 1,x 2, x 3, x 4,..., x n ) = c Calculus Overview Page 14