OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods.

1.1 Limits: A Numerical and Graphical Approach OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods.

1.1 Limits: A Numerical and Graphical Approach DEFINITION: As x approaches a, the limit of f (x) is L, written lim x a f (x) = L, if all values of f (x) are close to L for values of x that are sufficiently close, but not equal to, a. Slide 1.1-3

1.1 Limits: A Numerical and Graphical Approach THEOREM: As x approaches a, the limit of f (x) is L, if the limit from the left exists and the limit from the right exists and both limits are L. That is, if 1) lim f (x) = L, x a and then 2) lim f (x) = L, + x a lim x a f (x) = L, Slide 1.1-4

1.1 Limits: A Numerical and Graphical Approach The Wall Method: As an alternative approach to Example 1, we can draw a wall at x = 1, as shown in blue on the following graphs. We then follow the curve from left to right with pencil until we hit the wall and mark the location with an, assuming it can be determined. Then we follow the curve from right to left until we hit the wall and mark that location with an. If the locations are the same, we have a limit. Otherwise, the limit does not exist. Slide 1.1-14

1.2 Algebraic Limits and Continuity OBJECTIVES Develop and use the Limit Principles to calculate limits. Determine whether a function is continuous at a point.

1.2 Algebraic Limits and Continuity LIMIT PRINCIPLES: If lim f (x) = L and lim g(x) = M then x a x a we have the following: L.1 lim x a c = c The limit of a constant is the constant. Slide 1.2-3

1.2 Algebraic Limits and Continuity LIMIT PRINCIPLES (continued): L.2 The limit of a power is the power of the limit, and the limit of a root is the root of the limit. That is, for any positive integer n, and lim x a lim x a [ f (x)] n n = lim f (x) = L, x a n f (x) = n lim f (x) = L x a n, assuming that L 0 when n is even. Slide 1.2-4

1.2 Algebraic Limits and Continuity LIMIT PRINCIPLES (continued): L.3 The limit of a sum or difference is the sum or difference of the limits. lim x a [ f (x) ± g(x) ]= lim f (x) ± lim g(x) = L ± M. x a x a L.4 The limit of a product is the product of the limits. lim x a [ f (x)ig(x)]= lim f (x) x a i lim g(x) x a = LiM. Slide 1.2-5

1.2 Algebraic Limits and Continuity LIMIT PRINCIPLES (concluded): L.5 The limit of a quotient is the quotient of the limits. f (x) lim x a g(x) = lim f (x) x a lim g(x) = M L, L 0. x a L.6 The limit of a constant times a function is the constant times the limit. lim x a [ cf (x)]= cilim x a f (x) = cil. Slide 1.2-6

1.2 Algebraic Limits and Continuity THEOREM ON LIMITS OF RATIONAL FUNCTIONS For any rational function F, with a in the domain of F, lim x a F(x) = F(a). Slide 1.2-9

1.2 Algebraic Limits and Continuity DEFINITION: A function f is continuous at x = a if: 1) f (a) exists, (The output at a exists.) 2) lim f (x) exists, (The limit as x a exists.) x a 3) lim f (x) = f (a). (The limit is the same as the output.) x a A function is continuous over an interval if it is continuous at each point in that interval. Slide 1.2-13

1.3 Average Rates of Change OBJECTIVES Compute an average rate of change. Find a simplified difference quotient.

1.3 Average Rates of Change DEFINITION: The average rate of change of y with respect to x, as x changes from x 1 to x 2, is the ratio of the change in output to the change in input: y 2 y 1 x 2 x 1, where x 2 x 1. Slide 1.3-3

1.3 Average Rates of Change DEFINITION (concluded): If we look at a graph of the function, we see that y 2 y 1 = f (x 2 ) f (x 1 ), x 2 x 1 x 2 x 1 which is both the average rate of change and the slope of the line from P(x 1, y 1 ) to Q(x 2, y 2 ). The line through P and Q, PQ, is called a secant line. Slide 1.3-4

1.3 Average Rates of Change DEFINITION: The average rate of change of f with respect to x is also called the difference quotient. It is given by f (x + h) f (x), where h 0. h The difference quotient is equal to the slope of the line from (x, f (x)) to (x+h, f (x+h)). Slide 1.3-7

1.4 Differentiation Using Limits of Difference Quotients OBJECTIVES Find derivatives and values of derivatives Find equations of tangent lines

1.4 Differentiation Using Limits of Difference Quotients The slope of the tangent line at (x, f(x)) is m = lim h 0 f( x+ h) f x h This limit is also the instantaneous rate of change of f(x) at x. DEFINITION: (). Slide 1.4-3

1.4 Differentiation Using Limits of Difference Quotients DEFINITION: f For a function y = f (x), its derivative at x is the function f defined by ()= x lim h 0 f( x+ h) f x h () provided the limit exists. If f () x exists, then we say that f is differentiable at x. Slide 1.4-4

1.4 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable: 1) A function f(x) is not differentiable at a point x = a, if there is a corner at a. Slide 1.4-13

1.4 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable: 2) A function f (x) is not differentiable at a point x = a, if there is a vertical tangent at a. Slide 1.4-14

1.4 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable: 3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a. Example: g(x) is not continuous at 2, so g(x) is not differentiable at x = 2. Slide 1.4-15

1.5 Differentiation Techniques: The Power and Sum-Difference Rules OBJECTIVES Differentiate using the Power Rule or the Sum-Difference Rule. Differentiate a constant or a constant times a function. Determine points at which a tangent line has a specified slope.

1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules Leibniz s Notation: When y is a function of x, we will also designate the derivative, f () x, as dy dx, which is read the derivative of y with respect to x. Slide 1.5-3

1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules THEOREM 1: The Power Rule For any real number k, dy dx xk = k x k 1 Slide 1.5-4

1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules THEOREM 2: The derivative of a constant function is 0. That is, d dx c = 0 Slide 1.5-7

1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules THEOREM 3: The derivative of a constant times a function is the constant times the derivative of the function. That is, d dx c f (x) [ ] = c d dx f (x) Slide 1.5-8

1.5 Differentiation Techniques: The Power Rule and Sum-Difference Rules THEOREM 4: The Sum-Difference Rule Sum: The derivative of a sum is the sum of the derivatives. d [ f (x) + g(x) ] = d dx dx f (x) + d dx g(x) Difference: The derivative of a difference is the difference of the derivatives. d [ f (x) g(x) ] = d dx dx f (x) d dx g(x) Slide 1.5-11

1.6 Differentiation Techniques: The Product and Quotient Rules OBJECTIVES Differentiate using the Product and the Quotient Rules. Use the Quotient Rule to differentiate the average cost, revenue, and profit functions.

1.6 Differentiation Techniques: The Product and Quotient Rules THEOREM 5: The Product Rule Let F(x) = f (x) g(x). Then, F (x) = F (x) = d dx [ f (x) g(x) ] f (x) d dx g(x) + g(x) d dx f (x) Slide 1.6-3

1.6 Differentiation Techniques: The Product and Quotient Rules THEOREM 6: The Quotient Rule If Q(x) = N(x) then, D(x), Q (x) = D(x) N (x) N(x) D (x) [ D(x) ] 2 Slide 1.6-5

1.6 Differentiation Techniques: The Product and Quotient Rules DEFINITION: If C(x) is the cost of producing x items, then the C(x) average cost of producing x items is x. If R(x) is the revenue from the sale of x items, then the R(x) average revenue from selling x items is x. If P(x) is the profit from the sale of x items, then the P(x) average profit from selling x items is x. Slide 1.6-7

1.7 The Chain Rule OBJECTIVES Find the composition of two functions. Differentiate using the Extended Power Rule or the Chain Rule.

1.7 The Chain Rule THEOREM 7: The Extended Power Rule Suppose that g(x) is a differentiable function of x. Then, for any real number k, d dx g x () () k = k g x () k 1 d dx g x Slide 1.7-3

1.7 The Chain Rule DEFINITION: The composed function f og, the composition of f and g, is defined as f og = f (g(x)) Slide 1.7-6

1.7 The Chain Rule THEOREM 8: The Chain Rule The derivative of the composition f og is given by d dx ( f og)(x) [ ]= d dx f (g(x)) [ ]= f '(g(x)) g'(x) Slide 1.7-9