Math Problem Statement
Solution
Let's solve the problems displayed in the image one by one.
Problem 13:
Find an equation of the tangent line to the graph of at the specified point .
Given function:
- First, differentiate to find , which represents the slope of the tangent line at any point.
- Evaluate at to find the slope of the tangent line at .
- Use the point-slope form to find the equation of the tangent line: where and .
Problem 14:
Find an equation of the tangent line to the graph of the following function at the specified point .
Given function:
- Differentiate to find .
- Evaluate at to get the slope of the tangent line at that point.
- Use the point-slope form to find the equation of the tangent line, with the point and the calculated slope.
Problem 15:
Find the point(s) on the graph of where the tangent line is horizontal.
Given function:
- A horizontal tangent line means .
- Differentiate and set to find the critical points.
- Solve for the -values where this occurs.
- Evaluate at these -values to get the corresponding -values, giving you the points where the tangent is horizontal.
Problem 16:
Find the point(s) on the graph of the following function where the slope of the tangent line is equal to .
Given function:
- Differentiate to find .
- Set and solve for .
- Use the solved -values to find the corresponding -values by plugging them into .
Would you like the full solutions for these problems? Here are five follow-up questions to think about:
- What is the chain rule and how is it applied in differentiating the first function?
- How does one differentiate a quotient of two functions, as seen in Problem 14?
- What conditions are necessary for a tangent line to be horizontal?
- How do you solve for critical points where the slope is zero or equal to a specific value?
- What role does the second derivative play in analyzing the curvature of functions at these critical points?
Tip: Always check if a function is differentiable at the given point before attempting to find the tangent line.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Tangent Lines
Slopes
Critical Points
Formulas
Derivative formulas
Point-slope form of a line
Quotient rule
Product rule
Theorems
Differentiability
Tangent line approximation
Suitable Grade Level
College
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