Question

f(x)=2pix(1-x/g)^(1/7)

g=0.75

n=7

solve with trapezoidal rule and two point guess quadrant and also Analytically

Answer #1

(a) Use the Trapezoidal rule with 4 equal partitions to
approximate
?
integral (from -1 to 1) (x^2 +1)dx
via the formula Tn =(∆x/2)(y0+2y1+...+2yn−1+yn )with n=4, and
∆x=(b−a)/n
(b) Compare the actual error, found by direct integration minus
the approximation, with the known error bound for the Trapezoidal
rule
|ETn| ≤ (f′′(c)/12n^2) (b−a)^3, 12n2
where c is a point at which the absolute value of the second
derivative is maximized.

Use the trapezoidal rule with n=4 steps to estimate the
integral.
Integral from -1 to 1(x^2+6)dx
(-1 is on the bottom)

f(x)=x^3+1 and g(x) =x^-2
find rule for function
f-g

Consider the graph of y=f(x)=1−x2 and a
typical point P on the graph in the first quadrant. The tangent
line to the graph at P will determine a right triangle in the first
quadrant, as pictured below.
a) Find the formula for a function A(x) that computes the area
of the triangle through the point P=(x,y)
b) Find the point P so that the area of the triangle is as small
as possible: P =()

Using both the Midpoint and Trapezoidal Rules with n
subdivisions to approximate the integral of f(x) over the interval
from a to b numerically, where f(x) is concave up on [a,b], then we
would find?
Select one:
A. Midpoint overestimates, Trapezoid underestimates, Midpoint is
better
B. Midpoint underestimates, Trapezoid overestimates, Midpoint is
better
C. Midpoint overestimates, Trapezoid underestimates, Trapezoid
is better
D. Midpoint underestimates, Trapezoid overestimates, Trapezoid
is better
also,
If we used Gaussian Quadrature with 5 steps to approximate...

Find f o g and g o f
f(x)= 8/(x^2-1) , g(x) = x^3 +1
also find domain of f, g, f o g , and
g o f

Problem 1
f(x) = ex * cos(x)
Evaluate the integral of f(x) from x = - Π / 2 to x = + Π / 2
using:
1. Trapezoidal Rule where Δx = Π / 6
2. Simpson's 1/3 Rule where n = 4
3. Simpson's 3/8 Rule
4. Draw a scatter plot of f(x) in Excel over the desired range
(use sufficient data points to accurately depict the function)
5a. What is the true solution? (evaluate the integral)
5b....

(a) Suppose f and g are di?erentiable on an interval I and that
f(x) − ((g(x))n = c for all
xεI (where nεNand cεR are constants). If g(x) ̸= 0 on I, then
g′(x) = −f(x)((g(x))1−n.n
(b) If f is not di?erentiable at x0,then f is not continuous at
x0.
(c) Suppose f and g are di?erentiable on an interval I and
suppose that f′(x) = g′(x)on I.
Then f(x) = g(x) on I.
(d) The equation of the line...

Approximate the area under f(x) = (x – 2, above the
x-axis, on [2,4] with n = 4 rectangles using the (a) left endpoint,
(b) right endpoint and (c) trapezoidal rule (i.e. the “average”
shortcut). Be sure to include endpoint values and write summation
notation for (a) and (b). Also, on (c), state whether the answer
over- or underestimates the exact area and why.

1) Derive the quotient rule
( f g ) ′ = (f ′ ⋅ g − f ⋅ g ′ )/g 2
using the product rule and the chain rule.
I was able to get this with product rule(i think) but unsure how
to use chain rule
2) Identify a rate of accumulation (or decay) in your life
that changes over time. Be creative.
a) Describe it in a sentence or two: the rate of accumulation of
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