Answer to Question #108748 in Calculus for Preeti

Question #108748
Which of the following statements are true? Give reasons for your answers, in the
form of a short proof or a counterexample.
i)
2
2
2
dx
dy
dx
d y






=
ii) The inverse function of 3x
y = e is ln x
3
1
y = .
iii) If f is increasing and 0 )x(f > on an interval I, then
)x(f
1
)x(g = is decreasing on I.
iv) An equation of the tangent line to the parabola 2
y = x at )4,2 (− is
y − 4 = 2 x(x + )2 .
v) If f is one-one onto and differentiable on R , then
f )6(
1
f( )6()'
1

=

.
1
Expert's answer
2020-04-09T14:55:04-0400

1.


"{d^2y\\over dx^2}=({dy \\over dx})^2, False"

Let "y=x." Then


"{dy \\over dx}=1, {d^2y\\over dx^2}=0""{d^2y\\over dx^2}=({dy \\over dx})^2""{d^2y\\over dx^2}=0\\not=1=({dy \\over dx})^2"

2. The inverse function of "y=e^{3x}" is "y=\\dfrac{1}{3}\\ln x." False

If "x=-1: e^{3(-1)}=e^{-3}."

But the function "y=\\dfrac{1}{3}\\ln x" is undefined at "x=-1."

3.  If "f" is increasing and "f(x)>0" on an interval "I," then "g(x)=1\/f(x)" is deceasing on "I."

True

"f(x)>0" on I, "f(x)" is incresing on "I"

Let "x_1 \\in I, x_2\\in I"

"For \\ x_2>x_1, f(x_2)>f(x_1)>0"

Hence

"For \\ x_2>x_1, 0<1\/f(x_2)<1\/f(x_1)"

This means that "g(x)=1\/f(x)" is deceasing on "I."

4. An equation of the tangent line to the parabola "y=x^2" at "(-2,4)" is "y=2x(x+2)"

False. The equation of the line is "y=kx+b"


"k=y'(-2)=2(-2)=-4""y=-4x+b""y(-2)=-4(-2)+b=4=>b=-4"

The equation of the tangent line to th parabola "y=x^2" at "(-2, 4)" is


"y=-4x-4"

5.  If "f" is one - one onto and differentiable on R, then


"(f^{-1})'(6)=1\/f'(6)"

False

Let "y=x^3"

"y'=3x^2, y'(6)=3(6)^2=108"


"y^{-1}=\\sqrt[3]{x}"

"(y^{-1})'=\\dfrac{2}{\\sqrt[3]{x^2}}"

"(y^{-1})'(6)=\\dfrac{2}{\\sqrt[3]{6^2}}=\\dfrac{\\sqrt[3]{6}}{3}\\not=\\dfrac{1}{108}=\\dfrac{1}{y'(6)}"



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