/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Use \(f(t)=10 e^{-t}\). Evalua... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

Use \(f(t)=10 e^{-t}\). Evaluate \(f(0)\).

Short Answer

Expert verified
So, \(f(0)=10\).

Step by step solution

01

Express the Function

The function is given as \(f(t)=10 e^{-t}\).
02

Substitute \(t=0\) into function \(f(t)\)

After substituting, you get \(f(0) = 10 e^{-0}\).
03

Simplify

As \(e^0\) is equal to 1, the equation simplifies to \(f(0) = 10 \times 1\).
04

Get the final value

Multiplying through will give \(f(0) = 10\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
Function evaluation involves finding the output value of a function given an input value for the independent variable. This is an important aspect of understanding exponential functions and other types as well. When evaluating a function like \(f(t) = 10e^{-t}\), you replace the variable \(t\) with the given numerical value to find the result.

In this specific exercise, you need to evaluate the function at \(t = 0\). The purpose is to determine what the function's value is when the input is zero. Function evaluation helps us understand the behavior of functions and their specific outputs, which is critical in mathematical modeling and problem solving.
Substitution Method
The substitution method is a straightforward approach used in mathematics to replace variables in an equation or expression with given values. This technique simplifies the process of solving equations or evaluating functions by directly inserting the values into the predefined equation.

For the function given as \(f(t) = 10e^{-t}\), using the substitution method involves replacing \(t\) with \(0\). This step transforms the expression from \(10e^{-t}\) into \(10e^{-0}\), which can be subsequently evaluated. Substitution is an essential skill when it comes to working with functions, as it makes it possible to see exactly how functions behave with specific inputs.
Simplification Process
Simplifying an expression means breaking it down into the most basic and easily understandable form. The simplification process is crucial, especially when evaluating functions, as it allows you to determine the final, actual value of an expression.

Let's consider \(10e^{-0}\). According to exponential rules, any number raised to the power of zero is one. This applies to the constant \(e\), as \(e^0 = 1\). Therefore, \(10e^{-0}\) simplifies to \(10 \times 1\), culminating in the value \(10\).
  • This step not only emphasizes the power of exponential rules but also highlights the simplicity that can arise from understanding fundamental mathematical principles.
  • Through simplification, even seemingly complex expressions like exponential functions become manageable and intuitive.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The following table gives the price per barrel of crude oil for selected years from 1992 to 2006 (Source: www.ioga.com/special/crudeoil-Hist.htm) $$\begin{array}{|c|c|}\hline\text { Year } & \begin{array}{c}\text { Price } \\\\\text { (dollars) }\end{array} \\\\\hline 1992 & 19.25 \\\1996 & 20.46 \\\2000 & 27.40 \\\2004 & 37.41 \\\2006 & 58.30\\\ \hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(P(t)=C a^{t}\) that best fits the data. Let \(t\) be the number of years since 1992 (b) Using your model, what is the projected price per barrel of crude oil in \(2009 ?\)

Do all linear functions have inverses? Explain.

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 1500 ; r=6 \%\)

The function \(f(x)=|x+2|\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?

Give an example of a function that is its own inverse.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.