Graphical method for obtaining waveshape of current
Figure 1. Look, Mama, I rotated the picture! Graph of voltage vs time.

We'll need a little theory to help.

The capacitor

$$ Q = CV \tag 1 $$

Differentiating both sides gives us the rate of charge flow which is the current:

$$ I = \frac {dQ}{dt} = C \frac {dV}{dt} \tag 2 $$


We know from basic calculus that \$ \frac {dV}{dt} \$ is the slope of the line. If the slope is constant over a time period then the current must be constant during that period. 
The direction of the current will match the sign of the slope. Line sloping upwards means current in. Line sloping downwards means current out.
If the line is horizontal then \$ \frac {dV}{dt} = 0 \$ and therefore \$ I = 0 \$.


You should now be able to draw the curve for capacitor current.

The inductor

$$ V = L \frac {dI}{dt} \tag 3 $$


From (3) we can see that any time V is constant that \$ \frac {dI}{dt} \$ is constant. This means that for a constant non-zero voltage the current will continue to increase linearly with time. This part is counter-intuitive to some extent because in the back of our minds we remember that real inductors have internal series resistance. For this exercise we have to remember that ideal inductors have no internal resistance and so, with a constant voltage, the current can increase to infinity. That's the level part of the graph sorted out.


To be continued ...