Problem 26
Question
Graph \(e^{x}\) and \(\sqrt{e^{x}}\) at \(x=-2,-1,0,1,2 .\) Another form of \(\sqrt{e^{x}}\) is _______.
Step-by-Step Solution
Verified Answer
Another form of \(\sqrt{e^{x}}\) is \(e^{x/2}\).
1Step 1: Evaluate \(e^{x}\) at given points
Calculate the value of \(e^x\) for each given value of \(x\).- When \(x = -2\), \(e^{-2} \approx 0.1353\).- When \(x = -1\), \(e^{-1} \approx 0.3679\).- When \(x = 0\), \(e^{0} = 1\).- When \(x = 1\), \(e^{1} \approx 2.7183\).- When \(x = 2\), \(e^{2} \approx 7.3891\).
2Step 2: Graph \(e^{x}\)
Plot the points \((-2,0.1353), (-1,0.3679), (0,1), (1,2.7183), (2,7.3891)\) on a graph and draw the smooth exponential curve passing through these points. This represents the graph of \(e^x\).
3Step 3: Calculate and Graph \(\sqrt{e^{x}}\) at given points
For each \(x\), find \(\sqrt{e^x}\) and plot the points.- \(x = -2\), \(\sqrt{e^{-2}} \approx \sqrt{0.1353} \approx 0.3679\).- \(x = -1\), \(\sqrt{e^{-1}} \approx \sqrt{0.3679} \approx 0.6065\).- \(x = 0\), \(\sqrt{e^0} = \sqrt{1} = 1\).- \(x = 1\), \(\sqrt{e^1} \approx \sqrt{2.7183} \approx 1.6487\).- \(x = 2\), \(\sqrt{e^2} \approx \sqrt{7.3891} \approx 2.7183\).Graph the points \((-2,0.3679), (-1,0.6065), (0,1), (1,1.6487), (2,2.7183)\) and draw the square root curve.
4Step 4: Identify Alternative Expression
Recognize that another form of \(\sqrt{e^x}\) is expressed using exponents as \(e^{x/2}\). This is because taking the square root of an exponential function is equivalent to dividing the exponent by 2.
Key Concepts
GraphingSquare RootsExponents
Graphing
When you are graphing equations like \(e^x\) and \(\sqrt{e^x}\), it can feel like bringing a picture to life on a grid. Each point you calculate becomes a dot on your graph paper or screen, and connecting these points reveals the larger picture of the function. Let's break it down into simple steps.
You start by evaluating the function at various points. For \(e^x\), choose a series of \(x\)-values, such as \(-2, -1, 0, 1, 2\), and calculate the corresponding \(y\)-values. Once you have these points, plot them on a graph. The beauty of the exponential function \(e^x\) is in its rapid increase as \(x\) becomes larger. You'll notice the curve rises sharply to the right and flattens as \(x\) goes negative.
For \(\sqrt{e^x}\), the process is similar. By now, we know that taking the square root affects how steeply the graph climbs. Plot these points and draw a curve through them as well. Seeing these functions visually can make understanding their behavior much easier. Remember that
You start by evaluating the function at various points. For \(e^x\), choose a series of \(x\)-values, such as \(-2, -1, 0, 1, 2\), and calculate the corresponding \(y\)-values. Once you have these points, plot them on a graph. The beauty of the exponential function \(e^x\) is in its rapid increase as \(x\) becomes larger. You'll notice the curve rises sharply to the right and flattens as \(x\) goes negative.
For \(\sqrt{e^x}\), the process is similar. By now, we know that taking the square root affects how steeply the graph climbs. Plot these points and draw a curve through them as well. Seeing these functions visually can make understanding their behavior much easier. Remember that
- Each point corresponds to a coordinate \((x, y)\).
- The graph of \(e^x\) is steeper compared to \(\sqrt{e^x}\) due to square root damping the rate of increase.
- Graphs can help in predicting the values for non-integer \(x\).
Square Roots
Square roots are like a magic number that tells you when multiplied by itself, it gets you back to the original value you took it from. Let's tie this into our example of \(\sqrt{e^x}\). Understanding square roots can help us see why \(\sqrt{e^x}\) behaves the way it does.
When you have \(\sqrt{e^x}\), you're essentially finding a number that, when squared, would equal \(e^x\). Taking the square root of \(e^x\) is equivalent to using the expression \(e^{x/2}\). Why? Because the square root of an exponential function simplifies to half the exponent.
Here's the neat part:
When you have \(\sqrt{e^x}\), you're essentially finding a number that, when squared, would equal \(e^x\). Taking the square root of \(e^x\) is equivalent to using the expression \(e^{x/2}\). Why? Because the square root of an exponential function simplifies to half the exponent.
Here's the neat part:
- Think of square roots as questions of equality: "What number squared equals my original number?"
- Square roots reduce the steepness of an exponential rise, which is why \(\sqrt{e^x}\) isn't as steep as \(e^x\).
- They are a type of inverse operation, much like subtraction is to addition.
Exponents
Exponents are all about repeated multiplication. In the realm of exponential functions like \(e^x\), the exponent "\(x\)" determines how fast the function grows. It's a neat little number perched on top, pushing the base number \(e\) to grow or shrink.
In our exercises, \(e^x\) is the centerpiece. It describes how when you move to the right (the positive side of the \(x\)-axis), your function rises quickly, showing the explosive growth. Meanwhile, moving to the left (negative \(x\)-values) causes the function to approach zero, illustrating how powers of \(e\) behave when raised to negative numbers.
Some key points about exponents include:
In our exercises, \(e^x\) is the centerpiece. It describes how when you move to the right (the positive side of the \(x\)-axis), your function rises quickly, showing the explosive growth. Meanwhile, moving to the left (negative \(x\)-values) causes the function to approach zero, illustrating how powers of \(e\) behave when raised to negative numbers.
Some key points about exponents include:
- They indicate how many times to multiply the base number by itself.
- Negative exponents indicate division or "taking reciprocals."
- In functions like \(\sqrt{e^x}\), exponents assist in reformulating as \(e^{x/2}\), showing how they control the expression's scale or size.
Other exercises in this chapter
Problem 26
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which initial value produces \(y_{1}=y_{0}\) (steady state)? $$ y(t+1)=-\frac{1}{4} y(t)+6 $$
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