Problem 10
Question
Use the tangent line approximation of \(e^{x}\) at \(x=0\) to approximate \(e^{-1} .\) Is your answer larger than \(e^{-1}\) or smaller?
Step-by-Step Solution
Verified Answer
The tangent line approximation of \(e^{x}\) at \(x = 0\) to approximate \(e^{-1}\) is 0. This approximated answer is smaller than the exact value of \(e^{-1}\), which is approximately 0.368.
1Step 1: Calculate the derivative of the function
The derivative of the function \(e^x\) is also \(e^x\). At \(x = 0\), the derivative is \(e^0 = 1\).
2Step 2: Find the equation of the tangent line
The equation of a tangent line is \(y = f(a) + f'(a)(x - a)\). For the function \(e^x\) at \(x = 0\), this would be \(y = e^0 + 1(x - 0)\), which simplifies to \(y = 1 + x\).
3Step 3: Use the tangent line to approximate \(e^{-1}\)
Substitute \(x\) with \(-1\) in the tangent line equation to get an approximate value for \(e^{-1}\). So, \(y = 1 + (-1) = 0\).
4Step 4: Compare the approximation with exact value
The exact value of \(e^{-1}\) is approximately 0.368. The approximate value we obtained using the tangent line approximation is 0. Therefore, the approximated answer is smaller than the exact value of \(e^{-1}\).
Key Concepts
Derivative of Exponential FunctionEquation of Tangent LineApproximation ErrorsExponential Function Properties
Derivative of Exponential Function
The derivative of an exponential function helps us understand how the function changes at any given point. For the function \(e^x\), the derivative is remarkably simple: it is \(e^x\) itself. This means that at any point \(x\), the rate of change of the function is equal to its value at that point.
This is a unique property of the natural exponential function, making it especially useful in various mathematical contexts, including calculus and differential equations.
In the specific problem provided, we evaluated the derivative at \(x = 0\), leading to \(e^0 = 1\). This tells us that at \(x = 0\), the slope of the tangent line to the curve \(e^x\) is 1, providing a simple linear approximation to the curve near this point.
This is a unique property of the natural exponential function, making it especially useful in various mathematical contexts, including calculus and differential equations.
In the specific problem provided, we evaluated the derivative at \(x = 0\), leading to \(e^0 = 1\). This tells us that at \(x = 0\), the slope of the tangent line to the curve \(e^x\) is 1, providing a simple linear approximation to the curve near this point.
Equation of Tangent Line
The equation of a tangent line gives us a linear approximation to a curve at a specific point. It's especially useful when dealing with functions that may be complex or difficult to assess at a glance.
To find the equation of the tangent line, we use the formula \(y = f(a) + f'(a)(x - a)\). Here, \(f(a)\) is the function value at the point \(a\), and \(f'(a)\) is the slope of the tangent, which is the derivative.
In our example with \(e^x\) at \(x = 0\), the tangent line equation becomes \(y = 1 + 1(x - 0) = 1 + x\). This line approximates the behavior of the exponential function near \(x = 0\). By substituting \(x = -1\) into the equation, we can approximate the value of \(e^{-1}\), showing how such a linear equation can be applied practically.
To find the equation of the tangent line, we use the formula \(y = f(a) + f'(a)(x - a)\). Here, \(f(a)\) is the function value at the point \(a\), and \(f'(a)\) is the slope of the tangent, which is the derivative.
In our example with \(e^x\) at \(x = 0\), the tangent line equation becomes \(y = 1 + 1(x - 0) = 1 + x\). This line approximates the behavior of the exponential function near \(x = 0\). By substituting \(x = -1\) into the equation, we can approximate the value of \(e^{-1}\), showing how such a linear equation can be applied practically.
Approximation Errors
When we use a tangent line to approximate a function, we're engaging in a method known as the linear approximation. While often useful, this approach inherently includes some degree of error.
For accurate calculations, it's crucial to remember that the approximation is most reliable near the point where the tangent line is calculated. The further away from this point, the less accurate the approximation becomes.
In our exercise, approximating \(e^{-1}\) using the tangent line at \(x = 0\) resulted in an approximation of 0, while the true value is approximately 0.368. This discrepancy demonstrates the approximation error, which arises because the tangent line cannot perfectly match the curve at points further away from \(x = 0\).
For accurate calculations, it's crucial to remember that the approximation is most reliable near the point where the tangent line is calculated. The further away from this point, the less accurate the approximation becomes.
In our exercise, approximating \(e^{-1}\) using the tangent line at \(x = 0\) resulted in an approximation of 0, while the true value is approximately 0.368. This discrepancy demonstrates the approximation error, which arises because the tangent line cannot perfectly match the curve at points further away from \(x = 0\).
- This is a reminder of the limitations of linear approximations.
- It highlights how errors grow as we move away from the point of tangency.
Exponential Function Properties
Exponential functions, especially the natural exponential function \(e^x\), have fascinating properties that make them a cornerstone of higher mathematics. A defining feature is their constant relative rate of change.
The function \(e^x\) grows in such a way that at any point, its growth rate (derivative) is equal to its current value. This property is utilized in fields like population dynamics, interest calculations, and more complex calculus applications.
Moreover, the exponential function passes through the point (0, 1), and does not intercept the x-axis, which means it's never zero. It smoothly and continuously grows through positive and negative x-values, always remaining positive.
The function \(e^x\) grows in such a way that at any point, its growth rate (derivative) is equal to its current value. This property is utilized in fields like population dynamics, interest calculations, and more complex calculus applications.
Moreover, the exponential function passes through the point (0, 1), and does not intercept the x-axis, which means it's never zero. It smoothly and continuously grows through positive and negative x-values, always remaining positive.
- \(e^x\) retains its form even when differentiated.
- Its graph is always increasing, depicting continuous growth.
- It asymptotically approaches zero as \(x\) approaches negative infinity.
Other exercises in this chapter
Problem 9
Differentiate the function given. $$ f(x)=e^{2 x}\left(x^{2}+2 x+2\right) $$
View solution Problem 10
For Problems 10 through 15, factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{x+y}+b^{x} $$
View solution Problem 11
Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{2 x}+b^{x+1} $$
View solution Problem 11
Find the equation of the line tangent to \(f(x)=e^{x}\) at \(x=1\).
View solution