3. Consider the function f(x) 5 = 23 31 3° -x2 3 +13x+1 (a) Plot this function in Desmos. Use a viewing window with -2 ≤ x ≤ 6 and −14 ≤ y ≤ 19. On the same graph, plot the tangent line to y = f(x) at x = 1 and the tangent line at x = 3. (b) What will happen if you try to find a solution to f(x) = 0 with an initial guess of x0 = 1? Refer to your diagram to explain your answer. (Hint: Try doing the method for several iterations.) (c) Now use an initial guess of x0 = 5. Use Newton's method to find an approximate solution to f(x) = 0, by calculating 2. Give your answer to three decimal places. (d) Suggest an interval of values for the initial guess so that Newton's method will approach the middle x-intercept. Briefly explain your answer. [3 marks]

College Algebra
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ISBN:9781337282291
Author:Ron Larson
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Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
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Question
3. Consider the function
f(x)
5
=
23
31
3°
-x2
3
+13x+1
(a) Plot this function in Desmos. Use a viewing window with -2 ≤ x ≤ 6 and −14 ≤ y ≤ 19. On
the same graph, plot the tangent line to y = f(x) at x = 1 and the tangent line at x = 3.
(b) What will happen if you try to find a solution to f(x) = 0 with an initial guess of x0 = 1? Refer
to your diagram to explain your answer.
(Hint: Try doing the method for several iterations.)
(c) Now use an initial guess of x0 = 5. Use Newton's method to find an approximate solution to
f(x) = 0, by calculating 2. Give your answer to three decimal places.
(d) Suggest an interval of values for the initial guess so that Newton's method will approach the
middle x-intercept. Briefly explain your answer.
[3 marks]
Transcribed Image Text:3. Consider the function f(x) 5 = 23 31 3° -x2 3 +13x+1 (a) Plot this function in Desmos. Use a viewing window with -2 ≤ x ≤ 6 and −14 ≤ y ≤ 19. On the same graph, plot the tangent line to y = f(x) at x = 1 and the tangent line at x = 3. (b) What will happen if you try to find a solution to f(x) = 0 with an initial guess of x0 = 1? Refer to your diagram to explain your answer. (Hint: Try doing the method for several iterations.) (c) Now use an initial guess of x0 = 5. Use Newton's method to find an approximate solution to f(x) = 0, by calculating 2. Give your answer to three decimal places. (d) Suggest an interval of values for the initial guess so that Newton's method will approach the middle x-intercept. Briefly explain your answer. [3 marks]
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