Math Problem Statement
Solution
Let's break down the problem based on the image.
The problem asks you to:
- Find the equation of the secant line through the points where has the given values.
- Find the equation of the tangent line when has the first value.
The function provided is:
Given values for :
Step 1: Equation of the Secant Line
The slope of the secant line between two points and is calculated by:
First, let's calculate the function values at these points:
Now, the slope of the secant line:
The equation of the secant line is: Substitute , , and : Simplifying:
Step 2: Equation of the Tangent Line
The slope of the tangent line is the derivative of evaluated at .
The derivative of is:
Evaluating at :
The equation of the tangent line is: Substitute , , and : Simplifying:
Final Results:
- Secant line equation:
- Tangent line equation:
Would you like more details on this, or do you have any specific questions?
Here are 5 related questions to explore:
- How do we find the derivative of a function?
- Why is the slope of the tangent line different from the secant line?
- What are the geometric interpretations of secant and tangent lines?
- How would these equations change if the function were ?
- Can we find the secant line using another pair of points?
Tip: The tangent line always represents the instantaneous rate of change of the function at a specific point.
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Math Problem Analysis
Mathematical Concepts
Algebra
Calculus
Derivatives
Secant Line
Tangent Line
Formulas
Secant line slope formula: m_secant = (f(x2) - f(x1)) / (x2 - x1)
Equation of a line: y - y1 = m(x - x1)
Derivative of a function: f'(x)
Tangent line slope at a point: f'(x1)
Theorems
Definition of Derivative
Point-Slope Form of a Line
Suitable Grade Level
Grades 11-12 or early college
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