Problem 41 - Calculus: Graphical, Numerical, Algebraic

Question

Explorations In Exercises 4 and \(42,\) complete the following for the function. (a) Compute the difference quotient \(\frac{f(1+h)-f(1)}{h}\) (b) Use graphs and tables to estimate the limit of the difference quotient in part (a) as \(h \rightarrow 0\) . (c) Compare your estimate in part (b) with the given number. (d) Writing to Learn Based on your computations, do you think the graph of \(f\) has a tangent at \(x=1 ?\) If so, estimate its slope. If not, explain why not. \(f(x)=e^{x}, \quad e\)

Step-by-Step Solution

Answer

The limit of the difference quotient as \(h \rightarrow 0\) for the function \(f(x)=e^{x}\) is \(e\), and the slope of the tangent line to the function at \(x=1\) is also \(e\).

Step 1: Find the difference quotient

Firstly, we are asked to compute the difference quotient \(\frac{f(1+h)-f(1)}{h}\) for the function \(f(x)=e^{x}\). \n Substitute \(x=1\), \(x=1+h\) into the function and then plug them into the difference quotient formula: \n \(f(1+h) = e^{1+h} = e * e^{h}\), \n \(f(1) = e^{1} = e\), \n thus the difference quotient \(\frac{f(1+h)-f(1)}{h} = \frac{e * e^{h} - e}{h}\)

Step 2: Simplify the difference quotient

Simplify the difference quotient. Common factor \(e\) can get factored out of the numerator: \n\(\frac{e * e^{h} - e}{h} = \frac{e*(e^{h}-1)}{h}\)

Step 3: Estimate the limit of the difference quotient

We now estimate the limit of this difference quotient as \(h \rightarrow 0\). \nWe notice that, when \(h \rightarrow 0\), \(\frac{e * (e^{h}-1)}{h}\) trend towards \(e\). So the limit is \(e\).

Step 4: Compare and evaluate

Now compare this limit with the given number (unfortunately, the exercise does not supply a specific number for comparison). According to the question, the expectation would be that the estimated limit equals to the given number.

Step 5: Determine if the graph of the function has a tangent at \(x=1\)

Finally, determine whether the graph of \(f\) has a tangent at \(x=1\). Since the limit as \(h \rightarrow 0\) of the difference quotient exists, the function \(f(x)=e^{x}\) has a derivative at \(x=1\), and thus the graph of the function has a tangent at \(x=1\). The slope of that tangent is given by the calculated derivative, which is \(e\).

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at just one point. It's the best linear approximation of the curve at that particular point. For any differentiable function, the slope of the tangent line at a given point equals the derivative of the function at that point. Hence, a tangent line can help us understand the rate of change of a function.
In more practical terms:

• The tangent line gives a precise direction of the curve at a point.
• Its slope reflects how steep the curve is at that exact location.
• If the function is smooth (differentiable) at that point, the tangent line does exist.

In our exercise, we explored the function of an exponential curve, and by evaluating the derivative, we confirmed that the function does have a tangent line at \(x=1\). This tangent line's slope was found to be \(e\), or approximately 2.718. This is the key to differentiability: a smooth transition from the curve to the tangent.

Derivative Estimation

Estimating derivatives involves finding the slope of the tangent line at a specific point on a curve. The process commonly uses the difference quotient, \[ \frac{f(1+h)-f(1)}{h} \],where \(h\) is a small number. As \(h\) approaches 0, the difference quotient approaches the derivative of the function at \(x=1\).
This estimation solutions approach includes:

• Identifying how the function changes as \(h\) becomes smaller.
• Calculating values to see how they trend toward a limit.
• Recognizing that this limit, if it exists, represents the instantaneous rate of change of the function.

In our exercise, moving towards \(h=0\) allowed us to observe the limit of the difference quotient trending towards \(e\). This implies that the derivative, and hence the slope of the tangent line at \(x=1\), is indeed \(e\). It's a shining example of how visualizing and calculating these limits not only illustrates the derivative concept but confirms its existence and value.

Exponential Function

An exponential function is a type of mathematical function in the form \(f(x) = e^x\), where \(e\) is Euler's number, approximately 2.718. It is crucial due to its unique properties, which include rapid growth and constant proportional ratios.
When addressing the function \(f(x) = e^x\):

• Every point \(x\) has a corresponding \(y\) value calculated as \(e^x\).
• The function exhibits ongoing exponential growth, meaning it increases at increasing rates.
• The derivative of \(e^x\) is \(e^x\) itself, showcasing its growth potential.

This function's properties make it a keystone in both pure and applied mathematics, ranging from calculus to complex dynamic modeling in real-life contexts. When analyzing the derivative at a specific point like \(x=1\), we are uncovering the immediate rate of growth or decline at that exact point, providing us with insights into the nature of the function's behavior. It's these qualities that make the exponential function preferred for situations involving continuous growth.