Question

Define a function *f* as follows:

f ( x ) = sin ( x ) x , i f x ≠ 0 , a n d f ( 0 ) = 1.

Then *f* is a continuous function. Find the trapezoidal
approximation to the integral ∫ 0 π f ( x ) d xusing n = 4
trapezoids. Write out the sum formally and give a decimal value for
it.

Answer #1

Use Newton's method to find the absolute maximum value of the
function f(x) = 8x sin(x), 0 ≤ x ≤ π correct to
SIX decimal places.

Compute the area bounded by the function 50*sin(10*x^3-.5) and
the x-axis from x = π/6 to x = π/3 (Use 100 trapezoids). Write the
value below as the one displayed when you issue "format short" in
MATLAB.

Use the following table to estimate ∫0 25 f(x)dx. Assume that
f(x) is a decreasing function.
x
f(x)
0
50
5
46
10
42
15
35
20
29
25
9
To estimate the value of the integral we use the left-hand sum
approximation with Δx=
.
Then the left-hand sum approximation is ___ . To estimate the value
of the integral we can also use the right-hand sum approximation
with Δx=
Then the right-hand sum approximation is ____
.
The...

Consider the function f(x)=x⋅sin(x).
a) Find the area bound by y=f(x) and the x-axis over the interval,
0≤x≤π. (Do this without a calculator for practice and give the
exact answer.)
b) Let M(x) be the Maclaurin polynomial that consists of the
first 5 nonzero terms of the Maclaurin series for f(x). Find M(x)
by taking advantage of the fact that you already know the Maclaurin
series for sin x.
M(x)=
c) Since every Maclaurin polynomial is by definition centered at...

Let f(x) = sin(x) on the interval I = [0,π]. Also, let n =
10.
a.) Setting this problem up for a midpoint Riemann sum,
determine ∆x and a formula for
x∗k for the interval I and n given above:

Find the tangent to the function f(x) = sin(2x) at x = π.

(a) Find the Riemann sum for
f(x) = 3
sin(x), 0 ≤ x ≤
3π/2,
with six terms, taking the sample points to be right endpoints.
(Round your answers to six decimal places.)
R6 =
(b) Repeat part (a) with midpoints as the sample points.
M6 =
Express the limit as a definite integral on the given
interval.
lim n → ∞
n
7xi* +
(xi*)2
Δx, [3, 8]
i = 1
8
dx
3

Construct the Taylor series of f(x) = sin(x) centered at π.
Determine how many terms are needed to approximate sin(3) within
10^-9. Sum that many terms to make the approximation and compare
with the true (calculator) value of sin(3).

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

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