Math Problem Statement
Which is true in the photo
Solution
Let's analyze and solve the problems displayed in the uploaded images:
Image 1:
The task is to find the equation of the tangent line to the function at the point where .
Solution:
To find the equation of the tangent line to a function, we need two things:
- The slope of the tangent, which is the derivative of the function evaluated at .
- The point through which the tangent passes, which is .
Step 1: Derivative of : Now evaluate the derivative at : So, the slope of the tangent line is .
Step 2: Find the point on the curve at : Thus, the point on the curve is .
Step 3: Equation of the tangent line: The equation of a line is given by: Substitute , , and : Simplify: Thus, the equation of the tangent line is .
Correct answer: .
Image 2:
The task is to find the critical points of the function .
Solution:
Critical points occur where the derivative of the function is zero or undefined.
Step 1: Derivative of :
Step 2: Set the derivative equal to zero to find the critical points: Solve for :
So, there is one critical point at .
Step 3: Find the corresponding value of : Since we are only asked for the -values of the critical points, we can stop here.
Correct answer: , so the critical point is .
Let me know if you'd like further details or explanations!
5 Related Questions:
- How do you find the second derivative of a function, and what does it represent?
- What are inflection points, and how do you determine them?
- How do you find horizontal tangents of a function?
- What is the significance of the critical points in determining maxima and minima?
- How do you calculate the slope of a curve at a specific point?
Tip: Always simplify the derivative fully before solving for critical points or evaluating the slope!
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Math Problem Analysis
Mathematical Concepts
Derivative
Tangent Line
Quadratic Functions
Critical Points
Formulas
Derivative of a function: f'(x) = lim(h→0) [f(x+h) - f(x)]/h
Equation of a tangent line: y - y1 = m(x - x1)
Setting derivative to zero to find critical points: f'(x) = 0
Theorems
First Derivative Test for Critical Points
Slope of a tangent line
Suitable Grade Level
Grades 10-12
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