Textbook Question
Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
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4. Applications of Derivatives
Linearization
1:59 minutesProblem 4.6.7
Textbook Question
Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.
Verified step by step guidance1
Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point.
The formula for linear approximation is: \( L(x) = f(a) + f'(a)(x - a) \), where \( a \) is the point at which the function is known, \( f(a) \) is the function value at \( a \), and \( f'(a) \) is the derivative at \( a \).
In this problem, we are given \( f(4) = 3 \) and \( f'(4) = 2 \). We want to estimate \( f(3.85) \).
Substitute \( a = 4 \), \( f(a) = 3 \), \( f'(a) = 2 \), and \( x = 3.85 \) into the linear approximation formula: \( L(3.85) = 3 + 2(3.85 - 4) \).
Simplify the expression \( L(3.85) = 3 + 2(-0.15) \) to find the estimated value of \( f(3.85) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Approximation
Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that if a function is differentiable, its behavior can be closely approximated by a linear function in the vicinity of a specific input value.
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Tangent Line
The tangent line to a function at a given point is a straight line that touches the curve at that point and has the same slope as the function at that point. The equation of the tangent line can be expressed as y = f(a) + f'(a)(x - a), where 'a' is the point of tangency, f(a) is the function value, and f'(a) is the derivative at that point.
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Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the function's slope and is essential for finding the equation of the tangent line used in linear approximation.
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