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Chapter 3: Problem 75

Find the slope of the line tangent to the graph of \(f(x)=e^{x}\) at the point (0,1)

Short Answer

Expert verified
The slope of the tangent line at (0,1) is 1.

Step by step solution

01

- Understand the Function

The function given is \( f(x) = e^x \). This is an exponential function where the base is the mathematical constant \( e \).
02

- Identify the Derivative

The slope of the tangent line at any point on the curve can be found using the derivative. For the function \( f(x) = e^x \), the derivative \( f'(x) \) is also \( e^x \).
03

- Evaluate the Derivative at the Given Point

To find the slope of the tangent at the point \((0, 1)\), substitute \( x = 0 \) into the derivative: \( f'(0) = e^0 \).
04

- Calculate the Derivative

We know that \( e^0 = 1 \), so \( f'(0) = 1 \). Therefore, the slope of the line tangent to the graph of \( f(x) = e^x \) at the point \((0,1)\) is 1.

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Key Concepts

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

Exponential Function
An exponential function is a mathematical function of the form \( f(x) = a^x \), where the base \( a \) is a constant. In our problem, the function given is \( f(x) = e^x \), where the base is the special constant \( e \). This constant, known as Euler's number, is approximately equal to 2.71828. Exponential functions are essential because they model growth and decay processes, such as population growth, radioactive decay, and interest calculations. They have the property that their rate of change increases (or decreases) exponentially.
Derivative Calculation
To find the slope of the tangent line to the graph of a function at a given point, we first need to calculate the derivative. The derivative of a function \( f(x) \) represents the rate at which \( f(x) \) changes with respect to \( x \). For our specific function \( f(x) = e^x \), an amazing property is that the derivative of \( e^x \) is itself, \( e^x \). This means that regardless of the value of \( x \), the rate of change of \( e^x \) is always \( e^x \).
Evaluating Derivatives
Once we have the derivative, the next step is to evaluate it at the specific point we are interested in. In our problem, we need the slope of the tangent line at the point \((0, 1)\) on the graph of \( f(x) = e^x \). This involves substituting \( x = 0 \) into the derivative \( f'(x) = e^x \). So, we get \( f'(0) = e^0 \). Since any number raised to the power of 0 is 1, \( e^0 = 1 \). Therefore, the value of the derivative at \( x = 0 \) is 1.
Tangent Line
The tangent line to a curve at a given point is a straight line that just touches the curve at that point without crossing it. The slope of this tangent line is given by the value of the derivative at that point. In our case, we found that the derivative \( f'(x) \) at \( x = 0 \) is 1, so the slope of the tangent line to the graph of \( f(x) = e^x \) at the point \((0, 1)\) is 1. The tangent line equation can be written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Since the tangent line passes through the point \((0, 1)\), we have \( y = 1x + 1 \) or simply \( y = x + 1 \).

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Most popular questions from this chapter

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