{Use of Tech} Equations of tangent lines
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = e^x; a = ln 3

Verified step by step guidance

Identify the function and the point of tangency: The function is \( y = e^x \) and the point of tangency is at \( x = \ln 3 \).

Find the derivative of the function to determine the slope of the tangent line. The derivative of \( y = e^x \) is \( \frac{dy}{dx} = e^x \).

Evaluate the derivative at the point of tangency \( x = \ln 3 \) to find the slope of the tangent line. Substitute \( x = \ln 3 \) into the derivative: \( m = e^{\ln 3} \).

Calculate the y-coordinate of the point of tangency by substituting \( x = \ln 3 \) into the original function: \( y = e^{\ln 3} \).

Use the point-slope form of the equation of a line to write the equation of the tangent line. The point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Substitute the values obtained in the previous steps to get the equation of the tangent line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The slope of the tangent line can be found using the derivative of the function at that point. For the function y = e^x, the derivative is also e^x, which means the slope of the tangent line at any point is equal to the value of the function at that point.

Graphing Utilities

Graphing utilities are software or tools that allow users to visualize mathematical functions and their properties. They can plot curves, tangent lines, and other mathematical entities on the same axes, making it easier to analyze their relationships. In this context, a graphing utility can be used to graph the function y = e^x and the tangent line at the point where x = ln(3), providing a visual representation of the tangent's behavior.

Exponential Functions

Exponential functions are mathematical functions of the form y = a * b^x, where 'a' is a constant, 'b' is the base of the exponential, and 'x' is the exponent. The function y = e^x is a specific exponential function where 'e' is Euler's number, approximately equal to 2.71828. Exponential functions are characterized by their rapid growth and are commonly used in various fields, including calculus, to model growth processes.

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