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 a function can be closely approximated by a linear function when the input is near a specific value. The formula for linear approximation is f(a) + f'(a)(x - a), where f(a) is the function value at point a, 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 slope of the tangent line to the function at that point. For the function f(x) = √x, the derivative can be calculated using the power rule, which helps in determining the linear approximation.
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Choosing a Value for a
Selecting an appropriate value for 'a' is crucial in linear approximation to minimize error. Ideally, 'a' should be a value close to the point of interest (in this case, 146) where the function is easy to compute. For estimating √146, a nearby perfect square, such as 144 (where √144 = 12), can be chosen to ensure that the linear approximation yields a small error.
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Average Value of a Function
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