2. T2: I can generate an example and explain why that example satisfies or does not satisfy a collection of mathematical conditions. Theorem 1.1: Intermediate Value Theorem Suppose f is continuous on a closed interval [a, b]. If u is any number between f(a) and f(b), then there is at least one number c in ſa, b] such that f(c) = u. (a) Let g(x) = 1/x . Chose an interval where the Intermediate Value Theorem does not apply and explain why. Then, choose a u value for which there is no c where f(c) = u. (b) Let g(x) = 1/x . Chose an interval where the Intermediate Value Theorem does apply and explain why. (In other words, demonstrate why the hypothesis is satisfied.)

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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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2. T2: I can generate an example and explain why that example satisfies or does not satisfy a
collection of mathematical conditions.
Theorem 1.1: Intermediate Value Theorem
Suppose f is continuous on a closed interval [a, b]. If u is any number between f(a) and f(b), then
there is at least one number c in [a, b] such that f(c) = u.
(a) Let g(x) = 1/x . Chose an interval where the Intermediate Value Theorem does not apply
and explain why. Then, choose a u value for which there is no c where f(c)
= u.
(b) Let g(x) = 1/x . Chose an interval where the Intermediate Value Theorem does apply and
explain why. (In other words, demonstrate why the hypothesis is satisfied.)
3. T3: I can generate and use a counterexample to argue that a statement is false.
Generate a counter example (graph or formula) and use it to explain why the statement below is
false.
Suppose f is defined on a closed interval [a, b]. If u is any number between f(a) and f(b), then
there is at least one number c in [a, b] such that f(c) =
= u.
Transcribed Image Text:2. T2: I can generate an example and explain why that example satisfies or does not satisfy a collection of mathematical conditions. Theorem 1.1: Intermediate Value Theorem Suppose f is continuous on a closed interval [a, b]. If u is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = u. (a) Let g(x) = 1/x . Chose an interval where the Intermediate Value Theorem does not apply and explain why. Then, choose a u value for which there is no c where f(c) = u. (b) Let g(x) = 1/x . Chose an interval where the Intermediate Value Theorem does apply and explain why. (In other words, demonstrate why the hypothesis is satisfied.) 3. T3: I can generate and use a counterexample to argue that a statement is false. Generate a counter example (graph or formula) and use it to explain why the statement below is false. Suppose f is defined on a closed interval [a, b]. If u is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = = u.
4. L2: I can explain the purpose of each symbol in the definition of limit and illustrate each part of
the definition graphically.
Definition of Limit Let ƒ be a function defined on an interval around ¤ = c (not necessarily
at r = c.) We define the limit of the function f(x) as r approaches c, written lim f(x) to be the
number L (if one exists) such that f(x) is as close to L as we want whenever z is sufficiently close
to c (but r# c).
If L exists, we write lim f(x) = L.
(a) This question references the process done in this video: https://youtu.be/_bQNS3pwDrs
The graph of f(x) (below) is missing valuable information when r = 1! Let's see if we
can try to figure out what f(1) should be to fill in missing information. Follow the process
in the video to find an estimate of f(1) within 0.5 of the actual value. For I > 1 use
f(x) = -(x – 2)² + 3.8 and for a < 1, use f (x) = x + 1.8. Your final answer should be a
range of y values.
(d) N/)
(b) Label the following on the graph, using the numbers from (a) above. (Hint- all of these could
be located on the axes.)
i. L
ii. c
iii. "as close to L as we want" (As close to L
as you got in part (a))
iv. "sufficiently close to c" (use your part
(a))
4
Transcribed Image Text:4. L2: I can explain the purpose of each symbol in the definition of limit and illustrate each part of the definition graphically. Definition of Limit Let ƒ be a function defined on an interval around ¤ = c (not necessarily at r = c.) We define the limit of the function f(x) as r approaches c, written lim f(x) to be the number L (if one exists) such that f(x) is as close to L as we want whenever z is sufficiently close to c (but r# c). If L exists, we write lim f(x) = L. (a) This question references the process done in this video: https://youtu.be/_bQNS3pwDrs The graph of f(x) (below) is missing valuable information when r = 1! Let's see if we can try to figure out what f(1) should be to fill in missing information. Follow the process in the video to find an estimate of f(1) within 0.5 of the actual value. For I > 1 use f(x) = -(x – 2)² + 3.8 and for a < 1, use f (x) = x + 1.8. Your final answer should be a range of y values. (d) N/) (b) Label the following on the graph, using the numbers from (a) above. (Hint- all of these could be located on the axes.) i. L ii. c iii. "as close to L as we want" (As close to L as you got in part (a)) iv. "sufficiently close to c" (use your part (a)) 4
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