Answer to Question #344749 in Algebra for Geoffroy yawogan

Question #344749

1. Find the domain of the function 𝑓(π‘₯)=ln(βˆ’2π‘₯2βˆ’π‘₯βˆ’6)+√π‘₯2βˆ’1.

Β 

2. Find the inverse function of the function 𝑓(π‘₯)=π‘₯2βˆ’4π‘₯+5,π‘₯βˆˆγ€ˆ3,4⟩. Find the domain and the range of the inverse function.

Β 

3. Construct the tangent line to the graph of the function 𝑓(π‘₯)=4π‘₯β‹…βˆšπ‘₯βˆ’2β‹…βˆšπ‘₯ which is parallel to the line 𝑦=π‘₯.

Β 

4. Find the maximal intervals of monotonicity of the function 𝑓(π‘₯)=𝑒π‘₯+3π‘₯2+2π‘₯+6.

Β 

5. Find the integral ∫6β‹…π‘₯3⋅𝑒π‘₯2+2𝑑π‘₯.

Β 

6. Find the general solution of the differential equation π‘₯2+1+𝑦′⋅cos(𝑦)=0.Β 


1
Expert's answer
2022-05-26T07:03:25-0400

1.

a)


"f(x)=\\ln(-2x^2-x-6)+\\sqrt{x^2-1}"


"-2x^2-x-6>0""x^2-1\\ge0"


"D=(-1)^2-4(-2)(-6)=-47<0"

Then "-2x^2-x-6<0, x\\in \\R"


"Domain:\\{\\}"


b)


"f(x)=\\ln(-\\dfrac{2}{x^2-x-6})+\\sqrt{x^2-1}"


"x^2-x-6<0""x^2-1\\ge0"

"(x+2)(x-3)<0"

"x\\le -1\\ or\\ x\\ge1"

Domain:"(-2, -1]\\cup[1,3)"


2.


"\ud835\udc53(\ud835\udc65)=x^2\u22124x+5,x\\in [3, 4]"

"x_v=-\\dfrac{-4}{2(1)}=2"

The function "f" increases on "(3, 4)"


"f(3)=(3)^2\u22124(3)+5=2"

"f(4)=(4)^2\u22124(4)+5=5"

Domain: "[3, 4]"

Range: "[2, 5]"


"y=x^2-4x+5, 3\\le x\\le4"

ChangeΒ "x"Β andΒ "y"

"x=y^2-4y+5, 3\\le y\\le 4"

Solve forΒ "y"

"y^2-4y+4=x-1""(y-2)^2=x-1"

Since "3\\le y\\le 4"


"y-2=\\sqrt{x-1}"




"y=2+\\sqrt{x-1}"

Then


"f^{-1}(x)=2+\\sqrt{x-1}"


Domain: "[2, 5]"

Range: "[3, 4]"


3.


"f(x)=4x\\sqrt{x}-2\\sqrt{x}"

Domain: "[0, \\infin)"


"f'(x)=4(\\dfrac{3}{2})\\sqrt{x}-\\dfrac{2}{2\\sqrt{x}}=\\dfrac{6x-1}{\\sqrt{x}}"

"slope=f'(x)=\\dfrac{6x-1}{\\sqrt{x}}=1"

"6x-\\sqrt{x}-1=0"

"(3\\sqrt{x}+1)(2\\sqrt{x}-1)=0"

Since "\\sqrt{x}\\ge0," we take "2\\sqrt{x}-1=0"


"\\sqrt{x}=\\dfrac{1}{2}"

"x=\\dfrac{1}{4}"

"f(\\dfrac{1}{4})=4(\\dfrac{1}{4})\\sqrt{\\dfrac{1}{4}}-2\\sqrt{\\dfrac{1}{4}}=-\\dfrac{1}{2}"

The tangent line to the graph is


"y+\\dfrac{1}{2}=x-\\dfrac{1}{4}"


"y=x-\\dfrac{3}{4}"

4.


"f(x)=e^x+3x^2+2x+6"

Domain: "(-\\infin, \\infin)"


"f'(x)=e^x+6x+2"

"f'(x)=0=>e^x+6x+2=0"

"(e^x+6x+2)'=e^x+6>0, x\\in \\R"

The function "g(x)=e^x+6x+2" is strictly increasing on "(-\\infin, \\infin)."

The equation "e^x+6x+2=0" has the only solution "x\\approx-0.4406."


The function "f(x)" is monotonic decreasing on "(-\\infin, -0.4406)."

The function "f(x)" is monotonic increasing on "( -0.4406, \\infin)."


5.


"\\int 6x^3e^{x^2+2}dx"

"t=x^2, dt=2xdx"


"\\int 6x^3e^{x^2+2}dx=3e^2\\int te^tdt"

"\\int te^t dt"

"u=t, du=dt"

"dv=e^tdt, v=e^t"

"\\int te^t dt=te^t -\\int e^t dt=te^t-e^t+C_1"


"\\int 6x^3e^{x^2+2}dx=3e^{x^2+2}(x^2-1)+C"

Β 6.


"x^2+1+y'\\cos(y)=0"

"\\cos(y)dy=-(x^2+1)dx"

Integrate


"\\int \\cos(y)dy=-\\int (x^2+1)dx"

"\\sin (y)=-\\dfrac{x^3}{3}-x+C"

"\\sin (y)+\\dfrac{x^3}{3}+x=C"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS