Question

1. Find the domain of the function 𝑓(𝑥)=ln(−2𝑥2−𝑥−6)+√𝑥2−1.

 

2. Find the inverse function of the function 𝑓(𝑥)=𝑥2−4𝑥+5,𝑥∈〈3,4⟩. Find the domain and the range of the inverse function.

 

3. Construct the tangent line to the graph of the function 𝑓(𝑥)=4𝑥⋅√𝑥−2⋅√𝑥 which is parallel to the line 𝑦=𝑥.

 

4. Find the maximal intervals of monotonicity of the function 𝑓(𝑥)=𝑒𝑥+3𝑥2+2𝑥+6.

 

5. Find the integral ∫6⋅𝑥3⋅𝑒𝑥2+2𝑑𝑥.

 

6. Find the general solution of the differential equation 𝑥2+1+𝑦′⋅cos(𝑦)=0. 

Answer 1

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.

$$𝑓(𝑥)=x^2−4x+5,x\in [3, 4]$$

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

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

$$f(3)=(3)^2−4(3)+5=2$$

$$f(4)=(4)^2−4(4)+5=5$$

Domain: $[3, 4]$

Range: $[2, 5]$

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

Change $x$ and $y$

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

Solve for $y$

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

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.

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

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

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

$$\int te^t dt$$

$$u=t, du=dt$$

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

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

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

 6.

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

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

Integrate

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

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

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