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Chapter 4: Problem 23

Suppose \(t\) is a small positive number. Estimate the slope of the line containing the points \(\left(4, e^{4}\right)\) and \(\left(4+t, e^{4+t}\right)\)

Short Answer

Expert verified
The estimated slope of the line containing the points \((4, e^4)\) and \((4+t, e^{4+t})\) when t is a small positive number is approximately \(e^4\), which is the value of the derivative \(f'(4)\) for the function \(f(x) = e^x\).

Step by step solution

01

Identify the given points

The given points are (4,e^4) and (4+t,e^(4+t)).
02

Apply the slope formula

The slope (m) between our two points is given by: m = 鈭唝 / 鈭唜 where 鈭唝 = e^(4+t) - e^4 and 鈭唜 = (4+t) - 4
03

Simplify the slope expression

Now let's simplify the expressions for 鈭唝 and 鈭唜: 鈭唝 = e^(4+t) - e^4 鈭唜 = t So the slope, m, can be expressed as m = (e^(4+t) - e^4) / t
04

Consider t approaching zero

As t becomes smaller and smaller (approaching zero), the slope m will approach the value of the derivative of the function f(x) = e^x evaluated at x = 4. To find the derivative of f(x) = e^x, we see that: f'(x) = e^x Now, we can evaluate the derivative at x = 4: f'(4) = e^4
05

Estimate the slope and make the connection to the derivative

As t approaches zero, the slope of the line containing the points (4, e^4) and (4+t, e^(4+t)) will also approach the value of the derivative of the function f(x) = e^x, evaluated at x = 4. So we can estimate the slope to be approximately equal to the derivative f'(4) = e^4. Therefore, the slope of the line containing the points (4, e^4) and (4+t, e^(4+t)) is approximately e^4 when t is a small positive number.

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