SHOW ALL NEEDED WORK IN YOUR NOTEBOOK.

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1 DO NOW: 1 3: NO CALCULATORS 1. Consider the function f () x the value of f (4.1)? SHOW ALL NEEDED WORK IN YOUR NOTEBOOK. x. We all know that f (4), but without a calculator, what is

2 . The approximate value of y 4 sin() x at x 0.1, obtained from the tangent to the graph at x 0, is A..00 B..03 C..06 D..1 E..4

3 3. For small values of h, the function 4 16 h is best approximated by which of the following? A. 4 3 h B. 3 h C. h 3 h h D. 4 E. 3 3

4 4: CALCULATOR NEEDED 4. Let f be the function given by f () x x x 3. The tangent line to the graph of f at x is used to approximate the values of f () x. Which of the following is the greatest value for which the error resulting from this tangent line approximation is less than 0.5? A..4 B..5 C..6 D..7 E..8

5 DIFFERENTIALS Approximations aren t exact! (Aren t you glad you woke up this morning to hear that enlightening bit of information?!) If we use a line to approximate a curve, it gives us a good estimate, as long as we don t go too far away from the center point. Wouldn t be nice if we knew how far off our approximation is going to be? Well, whether you are excited about this or not, here we go!

6 Let s consider the function f(x) to the right, and the corresponding tangent line at the point with coordinates (c, f(c)). y If we want to approximate the value of f(x) for values of x close enough to x we may use the tangent line for such approximation. Equation of the Tangent Line at (c, f(c)) y f ()() c m x c ' y f ()()() c f c x c x y f ' ()()() c x c f c The tangent line is the graph is the graph of the Linear Function ' L()()()() x f c x c f c For as long as the line remains close to the graph of f, L(x) gives a good approximation of f(x). DEFINITION: Linearization If f is differentiable at x = c, then the approximating function ' L()()()() x f c x c f c Is the linearization of f(x) at x=c. Example #1: Find the linearization of f () x 1 x at x = 0.

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8 Example #: Find the linearization of f () x 1 x k at x = 0, for any constant k.

9 Example #3: Find the linearization of f () x cos() x at x

10 Example #4: Consider the function f below. Label the point c,() f c, and draw the tangent line at that point. y x a) What is the equation of the tangent line you drew? Keep in mind this is just a linearization of the curve. Move a small distance to the right of c. Normally, we would call this distance x is very small, we will instead use the notation dx, the differential of x. b) What is the function value at this point (when x c dx )? x, but when c) What is the value of this point on the tangent line (when x c dx )? d) How much did the y-values ACTUALLY change? e) How much did the y-values APPROXIMATELY change? (This is called dy the differential in y) In other words, if we were to use any value of x, the approximate change in y after a small change in x would be written This should look VERY familiar dy f ' () x dx

11 What differentials allow us to do is to say that if the ration of the differentials exists, it will be equal to the derivative. It allows us to write dy as the derivative of y with respect to x, but use the dy and dx dx as separate terms. Example #5: Find the differential dy if 3 y x 5x

12 Example #6: Find d cos(5) x.

13 Example #7: Since dy is the approximate change in the y-values when x is changed a small amount, we can use differentials to estimate the change in other problems if we know the small change in x. a) Find the differential dy when dx 0.01 and x, if y x x EXPLAIN what you ve found. b) Find the differential dy when dx 0. and x 1, if y x sin() x. EXPLAIN what you ve found. c) Without a calculator, use differentials to approximate 4..

14 NEWTON S METHOD Let s find a root for f x x 3 (between and 3) STEP 1 9 f (3) (3,1.5) ' f () x x ' f ( 3) 3 m 3 y 1.5 3( x 3) y 3x y 3x x 7. 5 Finding the y-int. for the tangent line, to be used as the new approximation for the root 7.5 x 3 x. 5 STEP 6.5 f (.5) (.5, 0.15) ' f () x x ' f (.5).5 m.5 y ( x.5) y.5x y.5x x 6.15 Finding the y-int. for the tangent line, to be used as the new approximation for the root 6.15 x.5 x.45

15 STEP f (.45) (.45, ) ' f () x x ' f (.45).45 m.45 y ( x.45) y.45x y.45x x Finding the y-int. for the tangent line, to be used as the new approximation for the root x.45 x

16 Newton s Method: x x f n 1 n f x x n n Example 8:

17 Example 9: Use the Newton s Method to solve x 3 3x 1 0.

18 CLASSWORK: V, where V 0 is the initial velocity in feet per 3 second and is the angle of elevation. If V 0 = 00 feet per second and changed from 10 to 11, use differentials to approximate the change in the range The range R of a projectile is R sin()

19 . The measurement of a side of a square is found to be 15 centimeters. The possible error in measuring the side is 0.05 centimeter. a) Approximate the percent error in computing the area of the square. b) Estimate the maximum allowable percent error in measuring the side if the error in computing the area cannot exceed.5%. c) Find the error.

20 3. A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less than 6%?

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