Consumer Awareness The cost C (in dollars) of making x...

Consumer Awareness The cost $C$ (in dollars) of making $x$ photocopies at a copy shop is given by $$ C(x)=\left\{\begin{array}{ll}{0.15 x,} & {0 < x \leq 25} \\ {0.10 x,} & {25 < x \leq 100} \\ {0.07 x,} & {100 < x \leq 500} \\ {0.05 x,} & {x > 500}\end{array}\right. $$ (a) Sketch a graph of the function. (b) Find each limit and interpret your result in the context of the situation. $$ \text { (i) }\lim _{x \rightarrow 15} C(x) \quad \text { (ii) } \lim _{x \rightarrow 99} C(x) \quad \text { (iii) } \lim _{x \rightarrow 305} C(x) $$ (c) Create a table of values to show numerically that each limit does not exist. $$ \text { (i) }\lim _{x \rightarrow 25} C(x) \quad \text { (ii) } \lim _{x \rightarrow 100} C(x) \quad \text { (iii) }_{x \rightarrow 500} C(x) $$ (d) Explain how to use the graph in part (a) to verify that the limits in part (c) do not exist.

Consumer Awareness The cost $C$ (in dollars) of
making $x$ photocopies at a copy shop is given by
$$
C(x)=\left\{\begin{array}{ll}{0.15 x,} & {0 < x \leq 25} \\ {0.10 x,} & {25 < x \leq 100} \\ {0.07 x,} & {100 < x \leq 500} \\ {0.05 x,} & {x > 500}\end{array}\right.
$$
(a) Sketch a graph of the function.
(b) Find each limit and interpret your result in the
context of the situation.
$$
\text { (i) }\lim _{x \rightarrow 15} C(x) \quad \text { (ii) } \lim _{x \rightarrow 99} C(x) \quad \text { (iii) } \lim _{x \rightarrow 305} C(x)
$$
(c) Create a table of values to show numerically that
each limit does not exist.
$$
\text { (i) }\lim _{x \rightarrow 25} C(x) \quad \text { (ii) } \lim _{x \rightarrow 100} C(x) \quad \text { (iii) }_{x \rightarrow 500} C(x)
$$
(d) Explain how to use the graph in part (a) to verify
that the limits in part (c) do not exist.

Key Concepts

Graphical Interpretation of Limits

Graphical interpretation involves using a graph to visually analyze the behavior of a function as the input nears a specific point. It aids in detecting discontinuities and understanding the overall trends. This method is essential for confirming analytical limit calculations, especially in cases where limits may not exist due to abrupt changes in the function's behavior.

One-sided Limits and Discontinuities

One-sided limits focus on the behavior of a function as the input approaches a specific point from one side (left or right). These are particularly important when dealing with discontinuities, such as jump discontinuities, where the approach from each side may yield different values. Recognizing one-sided limits allows for precise interpretation and explanation of discontinuities in piecewise-defined functions.

Numerical Verification of Limits

Numerical verification involves creating tables of function values that approach the target input from both sides. This technique complements analytical and graphical methods by providing concrete numerical evidence of the function’s behavior near points of interest. It is particularly useful for verifying the existence or non-existence of limits at points where a function may be discontinuous.

Piecewise Functions

Piecewise functions are those defined by different expressions over various intervals of the input variable. They are useful in modeling situations where a rule changes depending on the context, such as different pricing schemes at different production levels. Understanding how to evaluate and represent these functions is core to analyzing real-world scenarios and solving problems involving multiple conditions.

Limits

Limits examine the behavior of a function as its input approaches a specific value. This concept is foundational in calculus and is used to understand function behavior near points of interest, even where the function might not be defined. Limits help in analyzing trends and determining continuity of functions.

Transcript

00:01 All right, we want to talk about a piecewise function, and the piecewise function is c of x, and it will equal 0 .15 times x, but only for the x's, which are in between 0 and 25.

00:24 And notice it can equal 25, and this rule would apply.

00:28 Okay, it could also equal 0 .10 times x, and that would be for all the xes which are actually more than 25 but less than or equal to 100.

00:43 And then our next category is 0 .07 times x for all the xes, which are going to be more than 100, but less than or equal to 500.

01:03 And then our very last category is 0 .05 times x for all the the xes which are greater than 500.

01:17 Okay, so we have these four categories for x is between zero and going on to infinity and which rule we should apply.

01:29 Now, if we make a table for that, let's see if we can do that over here, if x is zero, the y would be 0 .15 times zero, and if x is 25, the y would be 0 .15 times zero.

01:46 And 5 times 25.

01:51 And those are all the x's that are, those are two examples of things from the first rule.

01:56 Now if we go to the second rule, let's say we go to 25.

02:00 And i know that the rule is really for only x's that are greater than 25, but if the rule had applied, it would be 2 .5.

02:10 And if it's at 100, the rule does apply, and it's 0 .10 times it, so that's 10.

02:18 And now we'll go to the 3 .3.

02:19 Rule, if it were 100, and i know the third rule doesn't apply when it's 100, but if it did, it would be at 7.

02:27 And if it's 500, it would be at 35.

02:34 And then last of all, last rule is anything that's actually over 500.

02:40 So the 500 doesn't apply on this one, but if it did, it would be at 25.

02:47 So the ones that apply are this one, this one, not that one, the 100 here, the 500 here.

02:55 But i want to keep the others because it helps me to make a good graph and to illustrate the point about what the limit would be.

03:02 So now, if i can actually draw the graph, let me erase this real quick.

03:15 And let me go back up a little bit.

03:21 We just want to come down.

03:22 Little.

03:25 The graph will look something like this.

03:32 And this is just going to be a real sketch of the graph.

03:36 If we have x is between 0 and 25, the rule is that it'll be 0 .15 times it.

03:43 So it'll be 0 to 0.

03:45 And then at 25, we're up to a value of 3 .75.

03:50 So we have a diagonal line looking something like this.

03:53 Now in the next category, which is 25 to 100, if it were in the next category, it would be at 2 .5, which would be right about here, open circle, not a closed circle.

04:10 So let me just kind of make sure that that's an open circle.

04:16 And then it'll be a diagonal line and closed circle here.

04:20 And then from 100 to 500, again, we have an open circle down here and then close circle up at the top.

04:33 And then above 500, we have an open circle down here and it will extend here.

04:41 So we have these broken diagonal looking lines.

04:44 And that's a rough idea of what it looks like.

04:47 Okay...

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