⧼exchistory⧽
6 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
For each of the following functions find those values of [math]x[/math] for which [math]f(x)[/math] or [math]f^\prime(x)[/math] vanish and use them to verify Rolle's Theorem.
- [math]f(x) = x^2 - 7x - 8[/math]
- [math]f(x) = 12x - x^3[/math]
- [math]f(x) = (x+4)(x+1)(x-2)[/math]
- [math]f(x) = x^2(x^2 - 16)[/math]
- [math]f(x) = x - \frac4x[/math]
- [math]f(x) = (9 - x^2)^2[/math].
BBy Bot
Nov 03'24
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[/math]
For each of the following functions and the specified values of [math]a[/math] and [math]b[/math], find a number [math]c[/math] such that [math]a \lt c \lt b[/math] and [math]f(b) = f(a) + (b-a)f^\prime(c)[/math].
- [math]f(x) = x^2 - 6x + 5, a = 1, b = 4[/math]
- [math]f(x) = x^3, a=0, b=1[/math]
- [math]f(x) = -\frac1x, a=1, b=3[/math]
- [math]f(x) = \frac8{x^2}, a=1, b=2[/math].
BBy Bot
Nov 03'24
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[/math]
Consider the function [math]f(x) = 1 - |x|[/math] defined on the closed interval from [math]-1[/math] to [math]1[/math]. Which hypotheses of Rolle's Theorem does this function satisfy and which does it not satisfy? Does this function satisfy the conclusion of Rolle's Theorem?
BBy Bot
Nov 03'24
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[/math]
Consider the function [math]f[/math] defined on the closed interval [math][4,7][/math] by
[[math]]
\dilemma{f(x)=0, & x=4,}{f(x)=7-x, & 4 \lt x \leq 7.}
[[/math]]
Show where this function fails to satisfy the conditions of Rolle's Theorem, and that is does not satisfy the conclusion.
BBy Bot
Nov 03'24
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[/math]
For each of the following functions [math]f[/math], find the set of all functions with derivative equal to [math]f[/math].
- [math]f(x) = 4x[/math]
- [math]f(x) = 4x^3 + x^2 + 2[/math]
- [math]f(x) = \frac1{x^2}[/math]
- [math]f(x) = \frac{2x}{(x^2 + 1)^3}[/math].
BBy Bot
Nov 03'24
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[/math]
Prove that, if on an automobile trip the average velocity was [math]45[/math] miles per hour, then at some instant during the trip the speedometer registered precisely [math]45[/math].