Consider the function (t) which is defined implicitly by the equation t+c= 1.) 1 √4+2 cos u @' (t) = 0" (t) = where c ER is a constant. a) Use upper and lower Riemann sums with 3 subintervals to estimate the value of c if 0(0) = π. b) State the Fundamental Theorem of Calculus I and use it to show that du, 4+2 cos(0(t)), sin(0(t)). c) Using the fact that 0(0) = π, show that the 3rd order Taylor polynomial of 0(t) about t = 0 is given by P3(t) = π + √√2t + 1 3√2 ) Give an approximate value of 0(1) using P3 (t) and use the Taylor remainder theorem to show that the error from the true value of 0(1) of the approximation is less than or equal to 7 4!
Consider the function (t) which is defined implicitly by the equation t+c= 1.) 1 √4+2 cos u @' (t) = 0" (t) = where c ER is a constant. a) Use upper and lower Riemann sums with 3 subintervals to estimate the value of c if 0(0) = π. b) State the Fundamental Theorem of Calculus I and use it to show that du, 4+2 cos(0(t)), sin(0(t)). c) Using the fact that 0(0) = π, show that the 3rd order Taylor polynomial of 0(t) about t = 0 is given by P3(t) = π + √√2t + 1 3√2 ) Give an approximate value of 0(1) using P3 (t) and use the Taylor remainder theorem to show that the error from the true value of 0(1) of the approximation is less than or equal to 7 4!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hi, I have a calculus question to ask. Thanks.
Transcribed Image Text:Consider the function (t) which is defined implicitly by the equation
t+c=
(1)
(t)=
0" (t) =
1
√4+2 cos u
where c ER is a constant.
a) Use upper and lower Riemann sums with 3 subintervals to estimate the value of c if 0(0)
b) State the Fundamental Theorem of Calculus I and use it to show that
du,
4+2 cos(0(t)),
sin(0(t)).
P3 (t) = x + √√√2t +
c) Using the fact that 0(0) = π, show that the 3rd order Taylor polynomial of 0(t) about t= 0 is
given by
<= π.
1 _t³.
3√/2
7
d) Give an approximate value of 0(1) using P3 (t) and use the Taylor remainder theorem to show
that the error from the true value of 0(1) of the approximation is less than or equal to
4!
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