Problem 10 - Algebra 2 and Trigonometry

Question

a. Make a table of values for \(e^{x}\) for integral values of \(x\) from \(-2\) to 3 b. Sketch the graph of \(\mathrm{f}(x)=e^{x}\) by plotting points and joining them with a smooth curve: c. From the graph, estimate the value of \(e^{\frac{1}{2}}\) and compare your answer to the value given by a calculator.

Step-by-Step Solution

Answer

Estimate \(e^{0.5}\) using the graph, then verify using a calculator (\(e^{0.5} \approx 1.649\)).

Step 1: Create a Table of Values

Begin by calculating the values of \(e^x\) for each integer \(x\) from -2 to 3. Use the formula \(y = e^x\) to obtain each result. The table will show values of \(x\) in one column and corresponding \(e^x\) values in the next.- For \(x = -2\), \(y = e^{-2} \approx 0.135\)- For \(x = -1\), \(y = e^{-1} \approx 0.368\)- For \(x = 0\), \(y = e^{0} = 1\)- For \(x = 1\), \(y = e^{1} \approx 2.718\)- For \(x = 2\), \(y = e^{2} \approx 7.389\)- For \(x = 3\), \(y = e^{3} \approx 20.086\)

Step 2: Sketch the Graph of f(x)

Using the table of values, plot the points on a Cartesian plane. Points to plot include: 1. \((-2, 0.135)\)2. \((-1, 0.368)\)3. \((0, 1)\)4. \((1, 2.718)\)5. \((2, 7.389)\)6. \((3, 20.086)\)Join these points smoothly to form the curve of \(f(x) = e^x\). Make sure the curve accurately represents the exponential growth as \(x\) increases.

Step 3: Estimate e^{1/2} Using the Graph

On the graph of \(f(x) = e^x\), locate the point where \(x = 0.5\) along the x-axis. Trace upwards to meet the curve and then horizontally to meet the y-axis. This point on the y-axis gives the value of \(e^{0.5}\). Estimate this value visually. The approximate value might be around 1.6 or 1.7.

Step 4: Compare with Calculator Value

Use a calculator to find the precise value of \(e^{0.5}\). The calculator gives \(e^{0.5} \approx 1.649\). Compare this with the estimated value from the graph obtained in the previous step. Notice that the estimate should be relatively close to the calculator value, confirming the accuracy of your graph.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept often observed in various natural and economic phenomena. It occurs when the rate of growth of a quantity is proportional to its current value. This means the larger the quantity gets, the faster it grows. This is quite different from linear growth, where the increase is constant regardless of size.

The function for exponential growth is usually written as \(f(x) = a \cdot b^x\), where \(a\) is the starting value, and \(b\) is the growth rate. When \(b > 1\), the function exhibits exponential growth. In the context of the exercise, we specifically look at \(f(x) = e^x\). This function is a prime example of exponential growth, where \(e\approx2.718\) is the base. With each increase in \(x\), the values of \(e^x\) increase rapidly, showcasing the unique nature of exponential processes. Understanding exponential growth helps us model and predict real-world occurrences like population growth, radioactive decay, and financial investments.

Graphing Functions

Graphing functions allows us to visualize how different mathematical rules apply across a range of inputs. When we graph \(f(x) = e^x\), we translate the algebraic expression into a visual picture on a Cartesian plane. This exercise involves plotting specific points calculated from the function on graph paper and drawing a smooth curve through these points.

• The x-axis (horizontal) represents the input values, while the y-axis (vertical) shows the corresponding output values of \(e^x\).
• Plot the points given in the exercise, such as \((-2, 0.135)\), \((0, 1)\), and \((3, 20.086)\).

Joining these points helps to form a curve that starts close to zero (for negative x-values) and rises steeply as x increases. The nature of the curve is a direct illustration of exponential growth, making the abstract concept more tangible.

Estimating Values from Graphs

Estimating values from graphs is a key skill that involves reading and interpreting data points visually. When you have a graph of a function, like \(f(x) = e^x\), you can find approximate values for points that aren't explicitly calculated.

Here's how to estimate \(e^{0.5}\) using the graph:
- Locate the point where \(x = 0.5\) on the x-axis.
- Draw a line vertically from this point until it intersects the curve of the function.
- From the intersection point, project horizontally to the y-axis.
The point where your horizontal line meets gives an estimated value of \(e^{0.5}\). In the exercise, this was around 1.6 or 1.7. By comparing this visually obtained estimate with the precise calculator value of 1.649, you can gauge the accuracy of your graph. Remember, estimating from graphs provides a useful approximation, which can be close but isn't exact.

Euler's Number

Euler's number, often denoted as \(e\), is a fundamental mathematical constant approximately equal to 2.71828. It's the base of natural logarithms and plays a crucial role in calculus, complex numbers, and mathematical modeling, particularly in exponential and logarithmic functions.

Named after the Swiss mathematician Leonhard Euler, \(e\) possesses numerous interesting properties.

• It appears in continuously compounding interest calculations, where it models the limit of compounding more frequently over a given period.
• In calculus, it arises when computing the derivative of \(f(x) = e^x\), where the derivative itself is \(e^x\), demonstrating its unique self-replicating attribute.

Thus, \(e\) not only assists in defining the natural logarithm but also serves as a backbone for various scientific disciplines, notably in growth and decay models. Recognizing Euler's number and its applications provides profound insights into mathematical and real-world systems.