Local Linear Approximation Objective: To estimate values using a local linear approximation.

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Presentation transcript:

Local Linear Approximation Objective: To estimate values using a local linear approximation

Zoom In If we use our calculator to magnify a portion of a graph, it looks linear. We will use that idea to approximate values. The value of the derivative can be looked at as the slope of the curve at a point.

Locally Linear A function that is differentiable at x is sometimes said to be locally linear at x. The line that best approximates the graph of f in the vicinity of P(x, f(x)) is the tangent line to the graph of f at x. We will use this equation:

Example 1 Find the local linear approximation of at Use the local linear approximation obtained to approximate.

Example 1 Find the local linear approximation of at Use the local linear approximation obtained to approximate. When x = 1, y = 1. The value of the derivative at x = 1 is so the local linear approximation is:

Example 1 Find the local linear approximation of at Use the local linear approximation obtained to approximate. We will use the equation to approximate

Example 1 Do you think that the approximation is bigger or smaller that the real value? If the graph is concave up around the point, the approximation is smaller than the real value and if the graph is concave down around the point, the approximation is larger than the real value.

Example 1 The derivative found in example 1 was Since we are interested in the concavity around 1 we need to take the second derivative and see if the value is positive (function is concave up) or negative (concave down).

Example 1 The derivative found in example 1 was Since we are interested in the concavity around 1 we need to take the second derivative and see if the value is positive (function is concave up) or negative (concave down). Since the second derivative is negative, the function is concave down around 1, so the approximation is larger.

Homework Section 3.5 3, 4