Math Problem Statement
Solution
Let's work through the problem step by step.
Problem:
We need to find the equations of the tangent lines to the curve at three points: , , and .
The general equation of the tangent line to a curve at a point is:
Where:
- is the slope of the tangent line at (which we get from the derivative of ),
- is the value of the function at .
1. Find the derivative of :
The derivative of is:
2. Calculate the slopes at each point:
At :
The slope of the tangent line is 0, so the line is horizontal.
The value of .
The equation of the tangent line is:
At :
The slope is 1.
The value of .
The equation of the tangent line is:
At :
The slope is -1.
The value of .
The equation of the tangent line is:
Summary of the Tangent Line Equations:
- At :
- At :
- At :
If you need more details or graphing explanations, let me know!
Here are five related questions:
- How is the derivative of derived, and what does it represent graphically?
- What happens to the tangent line when the slope is zero?
- How can we interpret the symmetry of the cosine function in relation to its tangent lines?
- Can you find the equation of a tangent line for at a specific point?
- How do we graphically determine the tangent line to a curve without calculating the derivative?
Tip: When finding the tangent line to a curve, always begin by calculating the derivative of the function to determine the slope at the desired point.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Calculus
Tangent Lines
Formulas
General equation of a tangent line: y = f'(a)(x - a) + f(a)
Derivative of y = cos(x): f'(x) = -sin(x)
Theorems
Differentiation of trigonometric functions
Tangent line theorem
Suitable Grade Level
Grades 10-12