
Linear Approximations
Flashcard
•
Mathematics
•
11th Grade
•
Hard
Wayground Content
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15 questions
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1.
FLASHCARD QUESTION
Front
What is the linear approximation of a function at a point?
Back
The linear approximation of a function at a point is the value of the function at that point plus the product of the derivative at that point and the change in x. It is given by the formula: \( L(x) = f(a) + f'(a)(x - a) \) where \( a \) is the point of approximation.
2.
FLASHCARD QUESTION
Front
How do you estimate \( \sqrt{23} \) using linear approximation at \( x = 25 \)?
Back
To estimate \( \sqrt{23} \) using linear approximation at \( x = 25 \), first find \( f(25) = 5 \) and \( f'(x) = \frac{1}{2\sqrt{x}} \) which gives \( f'(25) = \frac{1}{10} \). Then use the formula: \( L(23) = f(25) + f'(25)(23 - 25) = 5 - \frac{1}{10} \cdot 2 = 4.8 \).
3.
FLASHCARD QUESTION
Front
What is the formula for the derivative of a function?
Back
The derivative of a function \( f(x) \) at a point \( x \) is defined as: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
4.
FLASHCARD QUESTION
Front
How do you find the linear approximation of a differentiable function at a point?
Back
To find the linear approximation of a differentiable function \( f \) at a point \( a \), use the formula: \( L(x) = f(a) + f'(a)(x - a) \).
5.
FLASHCARD QUESTION
Front
What is the significance of the tangent line in linear approximation?
Back
The tangent line at a point on the graph of a function provides the best linear approximation of the function near that point.
6.
FLASHCARD QUESTION
Front
Given \( f(-2) = -4 \) and \( f'(x) = -2x - 1 \), how do you approximate \( f(-1.95) \)?
Back
Using the linear approximation: \( L(x) = f(-2) + f'(-2)(x + 2) \). Calculate \( f'(-2) = 3 \), then \( L(-1.95) = -4 + 3(0.05) = -3.85 \).
7.
FLASHCARD QUESTION
Front
What is the local linear approximation of a function?
Back
The local linear approximation of a function is the approximation of the function's value near a specific point using the tangent line at that point.
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