{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = e^x; a = ln 3

Verified step by step guidance

1. First, identify the function given in the problem. The function is \( y = e^x \).
2. To find the equation of the tangent line, we need the derivative of the function, which represents the slope of the tangent line at any point \( x \). The derivative of \( y = e^x \) is \( \frac{dy}{dx} = e^x \).
3. Evaluate the derivative at the given point \( a = \ln 3 \). Substitute \( x = \ln 3 \) into the derivative to find the slope of the tangent line: \( m = e^{\ln 3} \).
4. Calculate the y-coordinate of the point on the curve where \( x = \ln 3 \). Substitute \( x = \ln 3 \) into the original function: \( y = e^{\ln 3} \).
5. Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point on the curve and \( m \) is the slope. Substitute \( x_1 = \ln 3 \), \( y_1 = e^{\ln 3} \), and \( m = e^{\ln 3} \) to write the equation of the tangent line.

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative of the function.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function y = e^x, the derivative is also e^x, which simplifies the process of finding the slope of the tangent line.

Exponential Function

An exponential function is a mathematical function 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 case where the base 'e' is the natural logarithm base, and it has unique properties, such as its derivative being equal to itself, which is crucial for finding tangent lines.