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Rob
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Precalculus Functions Limits Derivative Evaluation ofderivatives
Theory
Rob
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Precalculus for 100.
lnx
y=
lnx+ ln ylnx− ln yx ln yy lnxnone of them
Rob
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Precalculus for 200.
The function y = x2 · sinx is
oddevenneither odd nor even
Rob
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Precalculus for 300.
arctan 1 =
∞π
3π
4π
6none of them
Rob
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Precalculus for 400.
The equivalence ”a < b if and only if f(a) < f(b)” isthe property of
even functionsone-to-one functionscontinuous functionsincreasing functionsnone of them
Rob
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Functions for 100.
How many points of inflection is on the graph of thefunction y = sinx in the open interval (0, 2π)
noneonetwothreenone of them
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Functions for 200.
Find points of discontinuity of the function y =x− 4
(x− 2) lnx
none00, 10, 1, 20, 20, 1, 40, 4none of them
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Functions for 300.
Let f be a function and f−1 be its inverse. Thenf−1
(f(x)
)=
01x
f(x)f−1(x)none of them
Rob
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Functions for 400.
arcsin(sinx) = x for every x ∈ R
YesNo
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Limits for 100.
limx→−∞
arctg x =
0π
2−π
2∞−∞none of them
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Limits for 200.
limx→∞
sinx =
1−1does not existnone of them
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Limits for 300.
limx→∞
2x3 + x2 + 4x2 − x+ 2
=
∞20none of them
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Limits for 400.
limx→0+
e1/x(x− 1)x
01e
∞−1−e−∞none of them
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Derivative for 100.(13√x
)′=
13x−2/3
−13x−2/3
−13x1/3
13x−4/3
−13x−4/3
none of them
Rob
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Derivative for 200.
(x− x lnx)′ =
lnx− lnx1 + lnx1− lnx0
1− 1x
none of them
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Derivative for 300.(x2ex2
)′2xe2x
2xex22x
2xex2+ x2ex2
2xex2+ x2ex2
2x2xex2
2x+ x2ex22x
none of them
Rob
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Derivative for 400.
The definition of the derivative of the function f atthe point a is
limh→0
f(x+ h) + f(x)h
limh→0
f(x+ h)h
limh→0
f(x+ h)− f(x)h
limh→0
f(x)− f(x+ h)h
limh→0
f(x− h)− f(x)h
none of them
Rob
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Evaluation of derivatives for 100.
(x2 + 1)′ =
Rob
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Evaluation of derivatives for 200.
(xex)′ =
Rob
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Evaluation of derivatives for 300.
ln(sinx) =
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Evaluation of derivatives for 400.
(xe−x)′ =
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Theory for 100.
By theorem of Bolzano, the polynomial y = x3+2x+4has zero on
(0, 1)(1, 2)(2, 3)(−1, 0)(−2,−1)(−3,−2)none of them
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Theory for 200.
Let a ∈ Im(f). Then the solution of the equationf(x) = a exists. This solution is unique if and only if
f is one-to-onef is increasingf continuousf differentiablenone of them
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Theory for 300.
If the function has a derivative at the point x = a,then it is
increasing at a.decreasing at a.one-to-one at a.continuous at a.undefined at a.
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Theory for 400.
If both y(a) = y′(a) = y′′(a) = 0, then the function
has local maximum at a.has local minimum at a.has point of inflection at a.any of these possibilites may be true, we needmore informations.
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