Problem 3 - Algebra and Trigonometry

Question

\(3-4\) . Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ h(x)=e^{x} ; \quad h(3), h(0.23), h(1), h(-2) $$

Step-by-Step Solution

Answer

h(3)≈20.085, h(0.23)≈1.259, h(1)≈2.718, h(-2)≈0.135

Step 1: Identify the Function and Values

We are given the function \( h(x) = e^x \) and need to evaluate it at specific values: 3, 0.23, 1, and -2. The task is to find \( h(3) \), \( h(0.23) \), \( h(1) \), and \( h(-2) \).

Step 2: Evaluate h(x) for x=3

Using a calculator, compute \( h(3) = e^3 \). This involves calculating the natural exponential function at 3. After computation, \( e^3 \approx 20.085 \).

Step 3: Evaluate h(x) for x=0.23

Next, compute \( h(0.23) = e^{0.23} \) using the calculator. The result, rounded to three decimal places, is approximately \( e^{0.23} \approx 1.259 \).

Step 4: Evaluate h(x) for x=1

Compute \( h(1) = e^1 \). Since \( e^1 = e \), the result is approximately \( e \approx 2.718 \).

Step 5: Evaluate h(x) for x=-2

Finally, compute \( h(-2) = e^{-2} \). Using a calculator, \( e^{-2} \approx 0.135 \) after rounding to three decimal places.

Key Concepts

Natural Exponent

The natural exponent is a fundamental concept in mathematics, particularly in calculus and exponential functions. The base of natural exponentials is a special irrational number known as Euler's number, denoted as \( e \). This number is approximately equal to 2.71828, but it extends infinitely without repeating. It is used as the base for natural logarithms and appears naturally in many phenomena involving growth and decay, such as population models and financial calculations.

When using the natural exponent in calculations, you're often looking at expressions in the form \( e^x \), where \( x \) represents a power or exponent. The function \( h(x) = e^x \) is an example of an exponential function where the base \( e \) is raised to the power of \( x \). This is called the natural exponential function. Understanding \( e \) and its properties is crucial because it describes many natural processes, sometimes making complicated phenomena more understandable.

Calculating Exponents

Calculating exponents means computing the value of a number raised to a given power. In the case of the natural exponent, you are raising the number \( e \) to the power \( x \). This can be done easily using a scientific calculator which typically includes an "\( e^x \)" or an "exp" button designed for this purpose.

Here’s a simple guide:

• Identify the value of \( x \) you are substituting into the function.
• Enter this value in the calculator after pressing "\( e^x \)".
• The calculator will give you the result, which is \( e \) raised to the power of \( x \).

Let's use this on a few examples:
For \( x = 3 \), calculating \( e^3 \) results in approximately 20.085. Similarly, for \( x = -2 \), \( e^{-2} \) is approximately 0.135.
When evaluating exponential functions, especially those with an irrational base like \( e \), a calculator is your best tool for quick and accurate calculations.

Rounding Decimals

Rounding decimals is an important technique to simplify complex numbers, especially when dealing with irrational numbers like \( e \). To round a number to three decimal places, you must first focus on the fourth decimal place: if it is 5 or greater, round the third decimal place up. If it's less than 5, leave the third decimal place unchanged.
Here's how you can approach it:

• Identify the number up to the third decimal place for accuracy.
• Check the fourth decimal place to decide whether to round up or down.
• For example, in \( e^3 \approx 20.08554 \), the fourth decimal place is 5, so we round up the third decimal place, resulting in 20.086.

Using this method will help ensure precision in your calculations, allowing you to faithfully represent the values when they're not exact. This is particularly useful in scientific contexts where significant figures are important for clarity and accuracy.