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Chapter 4: Problem 3

Find the slope of the tangent line to the exponential function at the point \((0,1) .\)

Short Answer

Expert verified
The slope of the tangent line to the exponential function at the point (0,1) is 1.

Step by step solution

01

Identify the Function

The exponential function is \(f(x) = e^x\). We are looking for the slope of the tangent line at the point (0,1).
02

Calculate the Derivative

To find the slope of the tangent line to the function at a certain point, we need to first find the derivative of the function. The derivative of \(f(x) = e^x\) is simply \(f'(x) = e^x\). This is because the derivative of \(e^x\) with respect to \(x\) is \(e^x\).
03

Find the Slope

To find the slope of the tangent line at the point (0,1), we substitute \(x = 0\) into the derivative we found in step 2. Therefore, the slope m = \(e^0 = 1 \).

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Most popular questions from this chapter

Use the given information to write an equation for \(y .\) Confirm your result analytically by showing that the function satisfies the equation \(d y / d t=C y .\) Does the function represent exponential growth or exponential decay? $$ \frac{d y}{d t}=-\frac{2}{3} y, \quad y=20 \text { when } t=0 $$

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