Differentials

# Differentials - Differentials Section 3.9 We used a tangent...

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Differentials Section 3.9

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We used a tangent line to approx. zeros in Newton’s Method. Now, we’ll use the tangent line to approx. the graph of a function on other situations. Ex. 1 pg. 221
Ex. 1 – Using a Tangent Line Approx. Find the tan. line approx. of f(x) =1+ sin x at (0,1) In other words, find the tangent line at (0,1) and then use the tangent line to approx. values of f(x). f ’(x) = cos x y - 1 = f ’(0) (x – 0) tangent line at (0,1) y – 1 = 1(x – 0) y = x + 1

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x -0.5 -0.1 -0.01 0 0.01 0.1 0.5 f(x) 1+sin x .521 .900 .9900 1 1.0099 1.0998 1.48 f ’(x) cos x .5 .9 .99 1 1.01 1.1 1.5 Notice that the closer x is to 0, the better the approximation is.
When the tangent line to the graph of f at the point (c,f(c)) : y = f(c) + f ’(c)(x-c) is used as an approx. to the graph of f , the quantity x-c is called the change in x or When is small, the change in y( ) can be approximated as follows: = f(c + ) – f (c) Actual change in y f ’(c) Approximate change in y Look at graph at top of pg. 222 y x x x x y

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Differentials - Differentials Section 3.9 We used a tangent...

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