Darboux's Theorem Prove Darboux’'s Theorem: Let f be differentiable on the closed interval [ a, b ] such that f ' ( a ) = y 1 , and f ' ( b ) = y 2 . If d lies between y 1 and y 1 , then there exists c in (a, b) such that f ' ( c ) = d .
Darboux's Theorem Prove Darboux’'s Theorem: Let f be differentiable on the closed interval [ a, b ] such that f ' ( a ) = y 1 , and f ' ( b ) = y 2 . If d lies between y 1 and y 1 , then there exists c in (a, b) such that f ' ( c ) = d .
Solution Summary: The author explains the Darboux's Theorem. The function h is continuous on left[a,bright] and has a relative minimum in this interval.
Darboux's Theorem Prove Darboux’'s Theorem: Let f be differentiable on the closed interval [a, b] such that
f
'
(
a
)
=
y
1
, and
f
'
(
b
)
=
y
2
. If d lies between
y
1
and
y
1
, then there exists c in (a, b) such that
f
'
(
c
)
=
d
.
Q1:
A slider in a machine moves along a fixed straight rod. Its
distance x cm along the rod is given below for various values of the time. Find the
velocity and acceleration of the slider when t = 0.3 seconds.
t(seconds)
x(cm)
0 0.1 0.2 0.3 0.4 0.5 0.6
30.13 31.62 32.87 33.64 33.95 33.81 33.24
Q2:
Using the Runge-Kutta method of fourth order, solve for y atr = 1.2,
From
dy_2xy +et
=
dx x²+xc*
Take h=0.2.
given x = 1, y = 0
Q3:Approximate the solution of the following equation
using finite difference method.
ly -(1-y=
y = x), y(1) = 2 and y(3) = −1
On the interval (1≤x≤3).(taking h=0.5).
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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