Andy Nguyen
Mrs Lomax
AP Calculus
October 13, 2017
Limit and continuity Journal Entry
Question 1: “Explain the importance of limits and continuity. Make sure to explain how limits and continuity are related. (Hints or topics to include: one sided limits, definition of continuity, important theorems, IVT)” The concept of the limit is one of the most essential things in order to prepare for Calculus. First, a limit is a certain number that one function can approach as an input comes closer and closer to one number. Limit can be found in many ways such as using substitution, graphical investigation, numerical approximation, algebra, or some combination of these. For example, the limit of the given function f(x) = 2x + 1 when x approaches 2 is …show more content…
A function y = f(x) is continuous point c if the limit of f(x) when x approaches c is equal to f(x). In other words, it is true only if the left and right limits are equal, and it also implies that the function exists at point c. Moreover, there is another way to check the continuity of one function by seeing whether the graph of a function can be traced with a pen without lifting the pen from the paper. According to the Intermediate Value Theorem, if K is continuous on the closed interval (a,b) then f takes on every value between f(a) and f(b). Suppose k is any number between f(a) and f(b), then there is at least one number c in (a,b) such that f(c) = k. For instance, let f(x) = X2+ XX-1, then verify that the Intermediate Value Theorem applies to the interval (2.5,4) and explain why the IVT guarantees an x-value of c where f(c) = …show more content…
The slope of a curve which is at the point P (a, f(a)) is the number m = h0f(a+h) - f(a)h. I will take an example to help people have clearer understanding about how do we use the slope of a curve. Let f(x) = x, write the equation of the tangent line when x = 9. First, we need to find the slope of the tangent line based on the equation above. h0f(a+h) - f(a)h h0a+h-ah h0(a+h-a) . a+h + hh . (a+h + h) h0a + h - a h . (a+h + h) h0 1a+h + h
After that, the limit of this function when h approaches 0 is , because I substitute h = 0 into the function. Then when x = 9, the slope of a curve is equal to 16. Substituting x = 9 into the equation then I will get f(x) = 3. Therefore, the equation of the tangent line when x = 9 is equal
Question 2: “What is the value of determining limits as x approaches infinity? For example in economic problems, the limit of a function can give important information. If an company has an increasing revenue or profit function, what does the limit as x approaches infinity tell us?” Extra
Mrs Lomax
AP Calculus
October 13, 2017
Limit and continuity Journal Entry
Question 1: “Explain the importance of limits and continuity. Make sure to explain how limits and continuity are related. (Hints or topics to include: one sided limits, definition of continuity, important theorems, IVT)” The concept of the limit is one of the most essential things in order to prepare for Calculus. First, a limit is a certain number that one function can approach as an input comes closer and closer to one number. Limit can be found in many ways such as using substitution, graphical investigation, numerical approximation, algebra, or some combination of these. For example, the limit of the given function f(x) = 2x + 1 when x approaches 2 is …show more content…
A function y = f(x) is continuous point c if the limit of f(x) when x approaches c is equal to f(x). In other words, it is true only if the left and right limits are equal, and it also implies that the function exists at point c. Moreover, there is another way to check the continuity of one function by seeing whether the graph of a function can be traced with a pen without lifting the pen from the paper. According to the Intermediate Value Theorem, if K is continuous on the closed interval (a,b) then f takes on every value between f(a) and f(b). Suppose k is any number between f(a) and f(b), then there is at least one number c in (a,b) such that f(c) = k. For instance, let f(x) = X2+ XX-1, then verify that the Intermediate Value Theorem applies to the interval (2.5,4) and explain why the IVT guarantees an x-value of c where f(c) = …show more content…
The slope of a curve which is at the point P (a, f(a)) is the number m = h0f(a+h) - f(a)h. I will take an example to help people have clearer understanding about how do we use the slope of a curve. Let f(x) = x, write the equation of the tangent line when x = 9. First, we need to find the slope of the tangent line based on the equation above. h0f(a+h) - f(a)h h0a+h-ah h0(a+h-a) . a+h + hh . (a+h + h) h0a + h - a h . (a+h + h) h0 1a+h + h
After that, the limit of this function when h approaches 0 is , because I substitute h = 0 into the function. Then when x = 9, the slope of a curve is equal to 16. Substituting x = 9 into the equation then I will get f(x) = 3. Therefore, the equation of the tangent line when x = 9 is equal
Question 2: “What is the value of determining limits as x approaches infinity? For example in economic problems, the limit of a function can give important information. If an company has an increasing revenue or profit function, what does the limit as x approaches infinity tell us?” Extra