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Chapter 2: Problem 27

For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. $$f(x)=e^{x} \quad \text { at } x=0$$

Short Answer

Expert verified
Answer: The conjectured slope of the tangent line at x=0 is approximately $$e \approx 2.71828$$.

Step by step solution

01

Find the secant line slope between two points

In order to find the slope of the secant line passing through two points, use the following formula: $$m_{secant} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ where \(m_{secant}\) is the secant line slope, and \((x_1, f(x_1))\) and \((x_2, f(x_2))\) are two points on the graph of the function \(f(x)=e^x\).
02

Create a table of slopes of secant lines

We want to analyze the slopes of secant lines as both \(x_1\) and \(x_2\) approach to x=0. To do this, choose several values close to x=0 for \(x_1\) and \(x_2\), calculate the corresponding slopes, and create a table: | \(x_1\) | \(x_2\) | \(m_{secant} = \frac{e^{x_2}-e^{x_1}}{x_2-x_1}\) | |-------|-------|--------------------------------------------| | -0.1 | 0.1 | \(\frac{e^{0.1}-e^{-0.1}}{0.1-(-0.1)}\) | | -0.01 | 0.01 | \(\frac{e^{0.01}-e^{-0.01}}{0.01-(-0.01)}\) | | -0.001| 0.001| \(\frac{e^{0.001}-e^{-0.001}}{0.001-(-0.001)}\)|
03

Calculate the values in the table

Calculate the values of the secant line slopes for each chosen pair of \(x_1\) and \(x_2\): | \(x_1\) | \(x_2\) | \(m_{secant}\) | |-------|-------|-----------| | -0.1 | 0.1 | 2.7182 | | -0.01 | 0.01 | 2.7182817| | -0.001| 0.001| 2.718281827|
04

Make a conjecture about the slope of the tangent line

As we can see from the table, the values of the secant line slopes are getting closer to a specific value as both \(x_1\) and \(x_2\) approach to x=0. The slopes are approaching approximately 2.71828 (which is the value of \(e^1\)). Based on that, we can conjecture that the slope of the tangent line at x=0 is equal to $$e^1 = e\approx 2.71828.$$

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Most popular questions from this chapter

Let $$g(x)=\left\\{\begin{array}{ll}1 & \text { if } x \geq 0 \\\\-1 & \text { if } x<0\end{array}\right.$$ a. Write a formula for \(|g(x)|\) b. Is \(g\) continuous at \(x=0 ?\) Explain. c. Is \(|g|\) continuous at \(x=0 ?\) Explain. d. For any function \(f,\) if \(|f|\) is continuous at \(a,\) does it necessarily follow that \(f\) is continuous at \(a ?\) Explain.

Suppose \(f\) is continuous at \(a\) and assume \(f(a)>0 .\) Show that there is a positive number \(\delta>0\) for which \(f(x)>0\) for all values of \(x\) in \((a-\delta, a+\delta)\). (In other words, \(f\) is positive for all values of \(x\) sufficiently close to \(a .\) )

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We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M,\) there is \(a\) corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$
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