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Math1011_Week04_Prelim1B

# Math1011_Week04_Prelim1B - Math1011 Spring 2005 Prelim Let...

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Math1011 Spring 2005 Prelim I-B1 Let f be the function graphed below: Is f continuous at 3? Justify your answer using the definition of continuity. At what numbers on is f discontinuous? x4 + What is lim ( )? fx - What is lim ( )? Note: stands for the set of Real Numbers. How would the answers differ if asked on the domain of f?

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Math1011 Spring 2005 Prelim B1 Let f be the function graphed below: Is f continuous at 3? Justify your answer using the definition of continuity. 3 3 Definition of Continuity: lim ( ) ( ) lim ( ) 0 but (3) 1 As lim ( ) (3), f(x) is not continuous at x = 3. xa x fx fa f = == At what numbers on is f discontinuous? x = 1 (jump discontinuity) x = 3 (removable discontinuity) x = 4 (infinite discontinuity) x4 + What is lim ( )? - - What is lim ( )? DNE (left sided limit does not equal right sided limit) How would the answers differ if asked on the domain of f? 4 is not on the domain of f and thus would not be included. (Doc #011p.14.01A)
Math1011 Spring 2002 Prelim B2 Draw the graph of a function y = f(x) such that the following conditions are satisfied: At x = 1: f (1) = 0 x1 lim ( ) 2 fx →+ = lim ( ) does not exist At x = 5: x5 lim ( ) 3 = f is not continuous at x = 5 At x = 9: slope of tangent line is 1 at x = 9 -4 -2 0 2 4 6 8 - 2 - 1 0123456789 1 0 1 1 1 2

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Math1011 Spring 2002 Prelim B2 Draw the graph of a function y = f(x) such that the following conditions are satisfied: At x = 1: f (1) = 0 x1 lim ( ) 2 fx →+ = lim ( ) does not exist At x = 5: x5 lim ( ) 3 = f is not continuous at x = 5 At x = 9: slope of tangent line is 1 at x = 9 (Doc #011p.26.03t)
Math1011 Fall 2002/Fall 2006 Prelim B3 53 Prove that there is a positive number c such that c2 1 0 cc −+ + = 2 If ( ) 10sin , show that there is a number c such that ( ) 1000. Give complete reasons for your answer. fx x fc =+ =

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Math1011 Fall 2002 Prelim B3 53 Prove that there is a positive number c such that c2 1 0 cc −+ + = Consider f(c) = c 2 f is a polynomial and thus continuous for all c. Let's try to find two values such that ( ) 10 ( ) f(1) = 1 1 1 2 3 f ( 2 ) = 2 2 22 3 2822 2 8 Intermediate Value Theorem As fa fb + << + = ++= −++= f is continuous on [1,2] and f(1) 10 (2), then there exists a number c in (1, 2) such that f( ) 10. So, there is a positive number c such that c 2 10. f c = + = 2 If ( ) 10sin , show that there is a number c such that ( ) 1000. Give complete reasons for your answer. fx x fc =+ = 2 2 f is continuous on (- , ) as it is the sum of a polynomial and a trig function. Let's try to find two values such that ( ) 1000 ( ) f(0) = 0 10sin(0) 0 0 0 1000 f(100) = (100) 10sin(100 ) 10,000 100 π ∞∞ += + = < > 0 Intermediate Value Theorem As f is continuous on [0,100] and f(0) 1000 (100), then there exists a number c in (0, 100) such that f( ) 1000. = (Doc #011p.26.04t)
Math1011 Fall 2002 Prelim B4 If ( ) 1, find the slope of f(x) at the point (3,2). fx x =+ Find an equation for the line tangent to f(x) at (3,2).

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Math1011_Week04_Prelim1B - Math1011 Spring 2005 Prelim Let...

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