Limits in Algebra and Calculus
Explore the concepts of limits in Algebra and Calculus, including practical applications and mathematical calculations. Understand how interest is compounded, domains of functions, and the geometrical meaning of limits of functions. Dive into the world of mathematical theory and practical value through engaging student activities.
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Presentation Transcript
Algebra & Calculus Limits
Is 2 euro the best possible return If I invest 1 1 after 1 year with an interest rate of 100%? http://ts3.mm.bing.net/th?id=H.4826792453277742pid=15.1H=160W=160
How often interest is compounded Final Value F Yearly = 2.0 Every 6 months = 2.25 Every 3 months = 2.44140625 Every month = 2.61303529 Every week = 2.69259695 Every day = 2.71456748 Every hour = 2.71812669 Every minute = 2.71827923 Every second = 2.71828162
2.7 2.71 2.718 2.7182 2.71828 2.718281 2.7182818 2.71828182 2.718281828 2.7182818284.............. n = e 1 n + lim 1 n
Assuming that the domains of the following functions are over R, write out the accurate domain for each function. ? ? ? ? = ? + 3 ? ? =?2 9 ? ?, ? 3 ? 3 1 ? = ? ?,? 3 ? 3
The practical value we use is called a limit and exists if the function approaches the same value from the left and right.
Concept of a limit Student Activity X 2.9 2.99 2.999 2.9999 3 3.0001 3.001 3.01 3.1 Page 3
Concept of a limit Student Activity X 2.9 2.99 2.999 2.9999 3 3.0001 3.001 3.01 3.1 5.9 5.99 5.999 5.9999 6 6.0001 6.001 6.01 6.1 5.9999 ? 6.0001 5.9 5.99 5.999 6.001 6.01 6.1 -10 -100 -1000 -10000 ? 10000 1000 100 10 6 6 No
Understanding the Geometrical Meaning of Limits of Functions
Understanding the Geometrical Meaning of Limits of Functions
Understanding the Geometrical Meaning of Limits of Functions
For a function to be continuous at a point it must fulfil all these 3 conditions: Are the above graphs continuous functions at x = 2?
Determine whether each of the following functions are continuous. If not state where the function is discontinuous Page 4
a) b) c) y = f(x) y = f(x) y = f(x) a a a d) e) f) a y = f(x) y = f(x) y = f(x) a a Page 5
