Problem 11 - Precalculus : Building Concepts and Connections

Question

use \(f(t)=10 e^{-t}\) Evaluate \(f(1)\)

Step-by-Step Solution

Answer

After simplification, \(f(1)\) is approximately \(3.678794\).

Step 1: Analyze the function

The function given is \(f(t)=10e^{-t}\), where \(f(t)\) is the function, \(t\) is the variable of the function and \(10e^{-t}\) is the expression which tells us how the function behaves. We are asked to find the value of the function when \(t = 1\).

Step 2: Substitute \(t = 1\) into the function

We plug in the value \(t = 1\) into the function: \(f(1) = 10e^{-1}\).

Step 3: Simplify the function

We simplify the above expression, where we have \(e^{-1}\) which means one over \(e\). So, \(f(1) = 10 \cdot \frac{1}{e}\). Since \(e\) has an approximate value of \(2.71828\), this can be simplified as \(f(1) = \frac{10}{2.71828}\).

Key Concepts

Exponential Decay

Exponential decay describes a process where the value of a quantity decreases at a rate proportional to its current value. This behavior is commonly found in nature and can be seen in the radioactive decay of atoms or the cooling of an object. The mathematical representation involves an exponential function with a negative exponent, as seen in the textbook exercise f(t) = 10e^{-t}.

In this example, as time t increases, f(t) decreases, exhibiting decay. Exponential decay models have a broad range of applications, from physics to finance, making understanding this concept crucial.

Natural Exponential Function

The natural exponential function is a special type of exponential function with base e, where e is an irrational number approximately equal to 2.71828. This base e is chosen for its unique properties, such as the rate of growth generated by e is the same as the current value.

In the textbook exercise, e^{-t} indicates a natural exponential function with a negative exponent which is key in modeling exponential decay. The natural exponential function is pivotal in calculus and appears in many natural phenomena, including growth and decay processes.

Function Evaluation

Function evaluation is a fundamental concept in mathematics involving the replacement of a function's variable with a specific value. The purpose is to compute the output of a function for a given input.

For example, evaluating f(1) where the function f(t) = 10e^{-t} involves replacing t with 1. This critical skill allows mathematicians and scientists to predict outcomes and analyze patterns efficiently.

Mathematical Simplification

Mathematical simplification is a process of reducing an expression into its simplest form. This makes it easier to understand and work with.

In the given textbook solution, simplification involves recognizing that e^{-1} is equivalent to 1/e. The further simplification using the approximate value of e assists in tangible interpretation and calculations in various exercises. Mastering simplification techniques is beneficial not only for clarity but also for solving complex problems more effectively.