Sketch these curves ay12x byelnx cy2x1 dy1lnxSolutiona y12x

Sketch these curves,

a)y=(-1/2)^x

b)y=e^lnx

c)y=2^|x+1|

d)y=1/lnx

Solution

a) y=(-1/2)^x is a function defined only for x=0 and positive integral powers of x. y values are oscillating about 0 , but the magnitude of the oscillations are progressively converging to zero as the integral values of x increases. The points of oscillations are in 1st and 4th quadrant.

b)y=e^lnx is equivalent to y=x which is a straight line with slope 45 degrees passing through the origin bisecting the first and 4th quadrant.

c)y=2^(x+1) is an exponential curve with base 2.

(-101, 1/2^100),.... (-1,1),(0,2), (1,4),(2,8), (3,(16)...(100, 2^101) are some of the points on the curve.

The curve intersects y axis at y=2.

y value is always positive.

The curve lies in 2nd and 1st quadrant.

As x-> -infinity, y approaches zero .

It is an increasing function.

y=0 or x axis is an asymptote.

d)y=1/lnx .

The function is not defined for x taking negative values.

For x=0, y=0. y is negative for x in the the interval [0 1[ but decreasing function. x=1 is excluded. At x=1, there is an infinite jump for y from minus infinity to +infinity. For x ]1 inf] y is again decreasing.

The curve is in the 4th quadrant for x in [0 1[ and in the 1st quadrant for x in]0 inf[

x=1 is an asymptote.

y=0 is an asymptote.