sol230921

Homework: Sketch graphs of the following functions. Please follow the steps I introduced at the lectures today. Please draw/sketch each graphs separately (one function, one graph). Also explicitly write down the equation(s) of the asymptotic line(s).

Solutions are given below.

(1) %5E2%7D%7B4(1-x)%7D#card=math&code=y%3D%5Cfrac%7B%281%2Bx%29%5E2%7D%7B4%281-x%29%7D&id=jUqYZ).

Solution: Domain . Not an odd/even/periodic function.

Taking derivative, we have

%5E2%7D%7B4(1-x)%5E2%7D%20%3D%20%5Cfrac%7B(x-3)(x%2B1)%7D%7B4(1-x)%5E2%7D.%0A#card=math&code=y%27%3D%20%5Cfrac%7B4-%281-x%29%5E2%7D%7B4%281-x%29%5E2%7D%20%3D%20%5Cfrac%7B%28x-3%29%28x%2B1%29%7D%7B4%281-x%29%5E2%7D.%0A&id=AI7RZ)

So that iff . And we have

	#card=math&code=%28-%5Cinfty%2C%20-1%29&id=mcTpg)		#card=math&code=%28-1%2C1%29&id=rYYO0)	#card=math&code=%281%2C3%29&id=vNBSF)		#card=math&code=%283%2C%20%5Cinfty%29&id=zKBBP)

Vertical asymptote: .

Special value: %3D0%2C%20y(0)%3D1%2F4#card=math&code=y%28-1%29%3D0%2C%20y%280%29%3D1%2F4&id=COJSA).

There is another asymptote for this function, whose equation is (x%2B3)#card=math&code=y%3D%28-1%2F4%29%28x%2B3%29&id=lw7mR). To compute it, note that

%5E2%7D%7B4x(1-x)%7D%3D-%5Cfrac%7B1%7D%7B4%7D%2C%20%5Cquad%20%5Clim%7Bx%20%5Cto%20%5Cinfty%20%7D%20%5B%5Cfrac%7B(1%2Bx)%5E2%7D%7B4(1-x)%7D-(-%5Cfrac%7B1%7D%7B4%7Dx)%5D%3D-%5Cfrac%7B3%7D%7B4%7D.%0A#card=math&code=%5Clim%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7B%281%2Bx%29%5E2%7D%7B4x%281-x%29%7D%3D-%5Cfrac%7B1%7D%7B4%7D%2C%20%5Cquad%20%5Clim_%7Bx%20%5Cto%20%5Cinfty%20%7D%20%5B%5Cfrac%7B%281%2Bx%29%5E2%7D%7B4%281-x%29%7D-%28-%5Cfrac%7B1%7D%7B4%7Dx%29%5D%3D-%5Cfrac%7B3%7D%7B4%7D.%0A&id=tiAni)

We will explain these two limits soon.

The graph of this function is

(2) #card=math&code=y%3Dx-%5Cln%281%2Bx%29&id=b7bCl) (here denotes the natural logarithmic function, that is, the base is e ).

Solution: Domain . Not an odd/even/periodic function.

Taking derivative, we have . So

	#card=math&code=%28-1%2C0%29&id=d5Tq7)		#card=math&code=%280%2C%20%5Cinfty%29&id=UUTxo)

Vertical asymptote: , because
No other asymptote. (The line is not an oblique asymptote for this function. See the graph below.)

Special value: %3D0#card=math&code=y%280%29%3D0&id=zMsJJ).

The graph is

(3) #card=math&code=y%3D%5Cln%281%2Bx%5E2%29&id=imGRF).

Solution: domain: all real . Even function. .

	#card=math&code=%28-%5Cinfty%2C%200%29&id=bAKDS)		#card=math&code=%280%2C%20%5Cinfty%29&id=tQAhf)

Again no asymptote. Special value: %3D0#card=math&code=y%280%29%3D0&id=TAfhZ).

The graph is

(4) .

Solution: domain . Odd function.

Taking derivative, we have

%5E2%7D%7B(1-x%5E2)-x(-2x)%7D%20%3D%20%5Cfrac%7Bx%5E2%2B1%7D%7B(1-x%5E2)%5E2%7D%20%5Cgeq%200.%0A#card=math&code=y%27%20%3D%20%5Cfrac%7B%281-x%5E2%29%5E2%7D%7B%281-x%5E2%29-x%28-2x%29%7D%20%3D%20%5Cfrac%7Bx%5E2%2B1%7D%7B%281-x%5E2%29%5E2%7D%20%5Cgeq%200.%0A&id=vzzVR)

So as increases, also increases.

Two vertical asymptotes: .

Horizontal asymptote: . This is because of .

No oblique asymptote.

Special values: %3D0%2C%20y(1%2F2)%3D2%2F3#card=math&code=y%280%29%3D0%2C%20y%281%2F2%29%3D2%2F3&id=EkJUM).

The graph is

(5) .

Solution: domain: all real . Even function. %20%5Cmathrm%7Be%7D%5E%7B-x%5E2%7D#card=math&code=y%27%3D%20-2x%28x%5E2-1%29%20%5Cmathrm%7Be%7D%5E%7B-x%5E2%7D&id=oMX4r).

	#card=math&code=%28-%5Cinfty%2C%20-1%29&id=M1Siz)		#card=math&code=%28-1%2C%200%29&id=gOwb2)		#card=math&code=%280%2C%201%29&id=zoM9g)		#card=math&code=%281%2C%20%5Cinfty%29&id=MQJgJ)

Horizontal asymptote: . This is because of

Special value: %3D0#card=math&code=y%280%29%3D0&id=nmrJm).

The graph is