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. \(f(x)=e^{x}, \quad e\)

Short 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-by-step solution

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 \(\frac{f(1+h)-f(1)}{h} = \frac{e * e^{h} - e}{h}\)

Simplify the difference quotient

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

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 \(e\).

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.

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 \(e\).