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Warm-Up/Activator Sketch a graph you would describe as continuous. Sketch a graph you would describe as discontinuous. Essential Question: What are the characteristics of a continuous function? Continuity Where am I continuous? Where am I discontinuous? x 5 x 12, ( x 4) 3 4, 5 x 2 f ( x) x2 , 2 x0 3 x 2, 0 x3 x 10, 3 x Definition of Continuity Let c be a number in the interval (a,b) and let f be a function whose domain contains the interval (a,b). The function f is continuous at the point c if the following conditions are true. 1. f(c) is defined 2. lim f ( x) exists x c f ( x ) f (c ) 3. lim x c Continuous Intervals If f is continuous at every point in the interval (a,b) then it is continuous on the interval (a,b) The domain of the function determines continuity. A polynomial function is continuous at every real number. A rational function is continuous at every number in its domain. Example 1] f(x) = x2 - 2x + 3 f(x) = x3 - x Example 2 Finding Discontinuities Determining Continuity of a Function A. f(x) = 1 x x 2 1 B. f(x) = x 1 1 C. f(x) = 2 x 1 Removable vs Non-removable x 3x 10 f ( x) 2 x 7 x 10 2 Holes are removable Vertical asymptotes (Infinite Discontinuities) and jump discontinuities are non-removable. Continuity on a Closed Interval Let f be defined on a closed interval [a,b]. If f is continuous on the open interval (a,b) and lim f ( x) f (a ) and xa lim f ( x) f (b) x b then f is continuous on the closed interval [a,b]. Moreover, f is continuous from the right at a and continuous from the left at b. Examining Continuity at Endpoints f ( x) 3 x Examining Continuity at Endpoints 5 x,1 x 2 f ( x) 2 x 1,2 x 3 Greatest Integer Function The Greatest Integer Function - is a step function x or [[x]] = greatest integer less than or equal to x Modeling a Cost Function A bookbinding company produces 10,000 books in an 8-hour shift. The fixed costs per shift amount to $5000, and the unit cost per book is $3. Using the greatest integer function, you can write the cost of producing x books as x 1 C 5000(1 [[ ]]) 3x 10000 Sketch the graph of this cost function Cost Function Graph Compound Interest Banks and other financial institutions differ on how interest is paid to an account. If the interest is added to the account so that future interest is paid on previously earned interest, then the interest is said to be compounded. Suppose, for example, that you deposit $10,000 in an account that pays 6% interest, compounded quarterly. Because the 6% is the annual interest rate, the quarterly rate is 1/4(.06) = 0.015 or 1.5%. Compound Interest Sketch the graph of the balance in the account described above. A = 10000(1+0.015)^[[4t]]