Question

In: Math

Consider the function f(x)=2e2sin(x). In this question, we will first use a linear approximation to estimate...

Consider the function f(x)=2e2sin(x). In this question, we will first use a linear approximation to estimate the value of f(0.1). Then, we will use a Taylor polynomial of degree three to estimate de value of f(0.1).

a) What is a good choice for the base point a of the linear approximation and the Taylor polynomial?

Answer: a=

b) Compute the derivative of f and evaluate it at x=a.

Answer: f′(x)=

f′(a)=

c) The linear approximation L(x) of f(x) based at a is:

Answer: L(x)=

d) Use the linear approximation that you have found in (c) to estimate f(0.1).

Answer: f(0.1)≈

e) Compute the second and third derivatives of f.

f′′(x)=

f′′′(x)=

f) Compute f′′(a) and f′′′(a).

f′′(a)=

f′′′(a)=

g) Find the Taylor polynomial of order three of f at the base point a.

T3(x)=

h) Use the Taylor polynomial that you found in (g) to estimate f(0.1).

Answer: f(0.1)≈

i) Compute the error (in absolute value) of the two approximations of f(0.1) that you have found. Give your answer with an accuracy of five decimal places.

Answer: Error for the linear approximation:

Error for the Taylor polynomial of order three:

To appreciate the improvement in the approximation provided by the Taylor polynomial of degree three over the linear approximation, you should sketch the graph of f, L, and T3 near a.

Expert Solution

milcah answered 2 years ago

1. Which of the following is the linear approximation of the function f ( x )...

1. Which of the following is the linear approximation of the function f ( x ) = 2e^sin (7x) at x = 0? Group of answer choices y=cos⁡(7)x+2 y=7x+2 y=14x+2 y=2x+7 y=e^7x+14 2. Recall that Rolle's Theorem begins, ``If f ( x ) is continuous on an interval [ a , b ] and differentiable on (a , b) and ___________, then there exists a number c …'' Find all values x = c that satisfy the conclusion of Rolle's...

Use finite approximation to estimate the area under the graph f(x)= 8x2 and above graph f(x)...

Use finite approximation to estimate the area under the graph f(x)= 8x2 and above graph f(x) = 0 from X0 = 0 to Xn = 16 using i) lower sum with two rectangles of equal width ii) lower sum with four rectangles of equal width iii) upper sum with two rectangles of equal width iv) upper sum with four rectangle of equal width

Please answer both parts of the question: a. Use linear approximation to estimate the value: 12.03/3.99...

Please answer both parts of the question: a. Use linear approximation to estimate the value: 12.03/3.99 b. Find ∂z/∂z: z3= x2sin(xyz)

Find the linear approximation of the function f(x,y)= e^(x^2 + 4xy - 2y^2) at (1,2) using the aproximate f(0.99,2.01)

-- Find the linear approximation of the function f(x,y)= e^(x^2 + 4xy - 2y^2) at (1,2) using the aproximate f(0.99,2.01) -- find Zvu for z= f(x,y), x=uv , y= v^2 + u^2 -

Question 1 Let f be a function for which the first derivative is f ' (x)...

Question 1 Let f be a function for which the first derivative is f ' (x) = 2x 2 - 5 / x2 for x > 0, f(1) = 7 and f(5) = 11. Show work for all question. a). Show that f satisfies the hypotheses of the Mean Value Theorem on [1, 5] b)Find the value(s) of c on (1, 5) that satisfyies the conclusion of the Mean Value Theorem. Question 2 Let f(x) = x 3 − 3x...

Consider the regression we know that f there is a strong linear correlation between X and...

Consider the regression we know that f there is a strong linear correlation between X and Z, then it is more likely that the t-test statistics get smaller. Show and explain how. Explain what happens if t-statistics get smaller.

Estimate the area (A) between the graph of the function F(X)=3/X and the interval [1,2]. Use an...

Estimate the area (A) between the graph of the function F(X)=3/X and the interval [1,2]. Use an approximation scheme with N= 2, 5 rectangles. Use the right endpoints. If your calculating utility will perform automatic summations, estimate the specified area using N=50 and N=100 rectangles. Round your answers to three decimal places. A2= A5= A10= A50= A100=

Consider the function f(x)= tan 1/x Use the following table to guess the limit f(x) as...

Consider the function f(x)= tan 1/x Use the following table to guess the limit f(x) as x goes to 0+ x f(x) = tan 1/x 1/π 1/2π 1/3π 1/4π 1/5π 4/π 4/5π 4/9π 4/13π 4/17π

Use matlab to plot Taylor approximation for f(x) for x near 0 Given f(x) = exp(x)...

Use matlab to plot Taylor approximation for f(x) for x near 0 Given f(x) = exp(x) Use subpots to plot and its absolute and realative erros for N= 1,2,3 PLease give matlab code for Taylor and explain in detail, Thank you

(a) Consider the function f(x)=(ex −1)/x. Use l’Hˆopital’s rule to show that lim f(x) = 1...

(a) Consider the function f(x)=(ex −1)/x. Use l’Hˆopital’s rule to show that lim f(x) = 1 when x approaches 0 (b) Check this result empirically by writing a program to compute f(x) for x = 10−k, k = 1,...,15. Do your results agree with theoretical expectations? Explain why. (c) Perform the experiment in part b again, this time using the mathematically equivalent formulation, f(x)=(ex −1)/log(ex), evaluated as indicated, with no simplification. If this works any better, can you explain why?...