# Tangent Lines and Logarithmic / Exponential Functions

Tangent Line and Logarithmic/Exponential Functions

This project is going to consider the relationship between a point (a, b) on the natural logarithmic and exponential functions and its relationship to the intercept of the tangent line to the respective functions at the point (a, b).

Hint: Recall the distance between two values a, b is

I. Natural Logarithm Function

1. Let L be the tangent line to at the point (1, 0) [See Figure 1]:

a. Find the equation of the line, L:

b. Let c be the y - intercept of the line L: What is the distance between c and 0?

2. Now, let T be the tangent line to at the point (a, b), where a, b are real numbers:

a. Find the equation of the line, T:

b. Let c be the y - intercept of the line T: What is the distance between c and b?

II. Exponential Function

3. Let L be the tangent line to at the point (0, 1) [See Figure 2]:

a. Find the equation of the line, L:

b. Let c be the x - intercept of the line L: What is the distance between c and 0?

4. Now, let T be the tangent line to at the point (a, b) where a, b are real numbers (Hint: ):

a. Find the equation of the line, T:

b. Let c be the x - intercept of the line T:

What is the distance between c and a?

III. Analysis:

In Part I of this project, you found that at any point (a, b) on the graph of , the distance between b (the y - coordinate) and the y - intercept of the tangent line to (a, b) is ____________.

In Part II of this project, you found that at any point (a, b) on the graph of , the distance between a (the x - coordinate) and the x - intercept of the tangent line to (a, b) is ____________.

In your own words, explain why these distances are the same.

(Hint: and share a special relationship)

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Calculus Project II:

Tangent Line and Logarithmic/Exponential Functions

This project is going to consider the relationship between a point (a, b) on the natural logarithmic and exponential functions and its relationship to the intercept of the tangent line to the respective functions at the point (a, b).

Hint: Recall the distance between two values a, b is

I. Natural Logarithm Function

1. Let L be the tangent line to at the point (1, 0) [See Figure 1]:

a. Find the equation of the line, L:

b. Let c be the y - intercept of the line L: what is the distance between c and 0?

, the slope of the tangent line at the point (1, 0) is

, thus the ...

#### Solution Summary

Tangent Lines and Logarithmic / Exponential Functions are investigated.