1.Find the domain of the function 𝑓(𝑥)=log4(𝑥2+4𝑥−5𝑒𝑥−1).
2. Construct the tangent line to the graph of the function 𝑓(𝑥)=√𝑥+2𝑥 which is perpendicular to the line 𝑥+3𝑦−2=0.
3. Find the maximal intervals of monotonicity of the function 𝑓(𝑥)=arctg(𝑥2−4𝑥).
4. Calculate the integral ∫𝑥2(𝑥3+4)2 𝑑𝑥2−1.
5. Find the particular solution of the differential equation 𝑦′=−(2𝑦+1)⋅tg(𝑥) which fulfills the initial condition 𝑦(𝜋4)=12.
6. Solve the matrix equation 2𝒳+𝒜=3ℬ−𝒜𝒳 if 𝒜=(−4−332), ℬ=(1−121).
1.
"x<-4.252"
Domain: "(-\\infin, -4.252)"
2.
"y=-\\dfrac{1}{3}x+\\dfrac{2}{3}"
The slope of the tangent line
"slope=m=-\\dfrac{1}{-\\dfrac{1}{3}}=3""\ud835\udc53(\ud835\udc65)=\\sqrt{x}+2\ud835\udc65"
"f'(x)=\\dfrac{1}{2\\sqrt{x}}+2"
"slope=m=3=\\dfrac{1}{2\\sqrt{x}}+2"
"\\dfrac{1}{2\\sqrt{x}}=1"
"x=\\dfrac{1}{4}"
"y(\\dfrac{1}{4})=\\sqrt{\\dfrac{1}{4}}+2(\\dfrac{1}{4})=1"
The equation of the tangent line in point-slope form is
The equation of the tangent line in slope-intercept form is
3.
Domain:"(-\\infin, \\infin)"
"f'(x)=0=>\\dfrac{2x-4}{1+(x^2-4x)^2}=0"
"x=2"
If "x<2, f'(x)<0, f(x)" decreases.
If "x>2, f'(x)>0, f(x)" increases.
The function "f" is monotone decreasing on "(-\\infin, 2)."
The function "f" is monotone increasing on "(2,\\infin)."
4.
"u=x^3+4, du=3x^2dx"
"\\int x^2(x^3+4)^2 dx=\\dfrac{1}{3}\\int u^2du=\\dfrac{u^3}{9}+C"
"=\\dfrac{(x^3+4)^3}{9}+C"
5.
Integrate
"\\int\\tan (x)dx=\\int\\dfrac{\\sin (x)}{\\cos(x)}dx=-\\ln|\\cos(x)|+\\dfrac{1}{2}\\ln C"
"\\dfrac{1}{2}\\ln|2y+1|=\\ln|\\cos(x)|+\\dfrac{1}{2}\\ln C"
"2y+1=C\\cos ^2 (x)"
Given "y(\\dfrac{\\pi}{4})=12"
"C=50"
The particular solution of the differential equation
"2y+1=50\\cos ^2 (x)"
6.
Let
Then
"AX=\\begin{pmatrix}\n -4 & -3 \\\\\n 3 & 2\n\\end{pmatrix}\\begin{pmatrix}\n x_{11}& x_{12} \\\\\n x_{21} & x_{22}\n\\end{pmatrix}"
"=\\begin{pmatrix}\n -4x_{11}-3x_{21}& -4x_{12}-3x_{22} \\\\\n 3x_{11}+2x_{21} & 3x_{12}+2x_{22}\n\\end{pmatrix}"
"3B-A=\\begin{pmatrix}\n 7 & 0 \\\\\n 3 & 1\n\\end{pmatrix}"
"\\begin{pmatrix}\n -2x_{11}-3x_{21}& -2x_{12}-3x_{22} \\\\\n 3x_{11}+4x_{21} & 3x_{12}+4x_{22}\n\\end{pmatrix}=\\begin{pmatrix}\n 7 & 0 \\\\\n 3 & 1\n\\end{pmatrix}"
"-2x_{12}-3x_{22}=0""3x_{12}+4x_{22}=1"
"x_{21}=27""-2x_{11}-3x_{21}=7"
"x_{11}=37, x_{21}=-27, x_{22}=-2, x_{12}=3"
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