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You are reading an older version of this FlexBook® textbook: CK-12 Texas Instruments Calculus Student Edition Go to the latest version.

This activity is intended to supplement Calculus, Chapter 1, Lesson 6.

Set up – graphing piecewise functions that show discontinuity

1. After turning on your device, go to the Y= screen by pressing [\blacklozenge] F1.

2. Turn the functions off or clear them; press F1 > Clear Functions.

Note: You can turn functions off by un-checking them using F4.

3. Turn Discontinuity Detection on. Press F1 > Format to find the option for Discontinuity Detection.

4. Set the window, using [\blacklozenge] F2, to the settings shown at the right.

5. Back on the Y= screen enter three piecewise functions.

At y1 press ENTER. Find when(in the CATALOG quickly by pressing CATALOG [ . ]. This shows the notation: when(condition, true, false)

For y1, type when(x<1,1,a)|a=5

The “such that” bar key ([ | ]) is to the left of the [7] key.

y1(x)= \begin{cases}   1,x<1 \\\qquad \quad \ |a=5 \\  a,x \ge 1 \end{cases}

For y2, type when(x<1,x+2,a^*x^{\land}2)|a=5

y2(x)=         \begin{cases}   x+2, x < 1 \\\qquad \qquad \quad |a=5 \\  a \cdot x^2, x \ge 1 \end{cases}

For y3, type when (x < 2,2 \sin \left((x - 1) \frac {\pi}{2} \right),a + 3 \sin \left((x-4)\frac {\pi}{2} \right)|a=5

y3(x)= \begin{cases}   2 \sin \left ((x-1) \frac{\pi}{2} \right ) ,x<2 \\\qquad \qquad \qquad \qquad \qquad |a=5 \\  a +  3 \sin \left ((x-4) \frac {\pi}{2} \right ),x \ge 12\end{cases}

6. Graph one function at a time by using F4 to have only one function checked at a time.

On a graph screen examine both sides of where the discontinuity exists using F3 Trace.

7. For Problems 1 and 2 below, use [\blacklozenge] F4 to have table settings of \text{tblStart} = 0.98 and \triangle \text{tbl}= 0.01, to numerically examine the left and right-hand limits. Be sure to press ENTER to save changes before pressing [\blacklozenge] F5 to view the table.

For Problems 1, 2, and 3 estimate the limits graphically and numerically using trace and table.

Problem 1

y1(x) =           \begin{cases}   1,x<1 \\\qquad \qquad |a=5 \\  a,x\ge1 \end{cases}

\lim_{x \to 1^-}y1(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}

\lim_{x \to 1^+}y1(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}

Try other values for a in the graph of y1(x) to find what a makes \lim_{x \to 1}y1(x) exist. On the Y= screen, press ENTER when y1 is highlighted. Press  \blacktriangleright and then backspace \leftarrow to try different values for a. Graph it to see if appear continuous.

a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}

Problem 2

y2(x)=           \begin{cases}   x+2, x<1 \\\qquad \qquad \quad|a=5 \\  a \cdot x^2,x\ge1 \end{cases}

\lim_{x \to 1^-}y2(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}

\lim_{x \to 1^+}y2(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}

Try other values for a in the graph of y2(x) to find what a makes \lim_{x \to 1}y2(x) exist.

a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}

Show calculations of the left hand limit and the right hand limit to verify that your value for a makes the limit exist.

Problem 3

y3(x)=           \begin{cases}   2\sin \left((x-1)\frac{\pi}{2} \right),x<2 \\\qquad \qquad \qquad \qquad \qquad |a=5 \\  a+3\sin \left((x-4)\frac{\pi}{2} \right),x\ge2 \end{cases}

\lim_{x \to 2^-} y3(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}

\lim_{x \to 2^+}y3(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}

Try other values for a in the graph of y3(x) to find what a makes \lim_{x  \to 2} y3(x) exist.

a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}

Show calculations of the left hand limit and the right hand limit to verify that your value for a makes the limit exist.

Extension – Continuity

A function is continuous at x=c if:

  • f(c) exists
  • \lim_{x \to c}f(x) exists, and
  • \lim_{x \to c}f(x)=f(c)

Use CAS to algebraically solve for a that makes

(a) \lim_{x \to 1}y2(x) exist

(b) \lim_{x \to 2}y3(x) exist

Then prove each function is continuous.

Key press help:

  • Begin by pressing HOME. Clean Up the screen by pressing 2^{nd} [F1]. Choose NewProb and press ENTER to put this on the command line and ENTER to execute the command.
  • Type y2(x) ENTER. The Define command is under the F4 menu. Type Define f(x)=, then up arrow to highlight the output from the previous line. Press ENTER on the highlighted piecewise function to copy it down to the command line.

  • To solve a right sided limit, press F3 > limit(. On the command line enter limit(f(x),x,1,1) ENTER.
  • Now, press F2 ENTER to select solve( Then up arrow to select the input from the previous line, press ENTER. Next type [ -- ]. Up arrow to the input again and press ENTER. This time put a negative (-) in front of the last 1. Finally type [ , ] ALPHA [ -- ] and close the parentheses. This method will enable you to quickly enter solve(limit(f(x),x,1,1) = \text{limit}\ (f(x) ,x, 1,-1),a).

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