Curve Sketching using derivative properties

Make a careful sketch of a smooth curve (if possible) having all the properties stated. If a smooth curve can not be sketch, then make sure it’s continuous

(no breaks).

f(0)=0, f ′ (x) › 0 for 0 ‹ x ‹ 1, f ′ (x) = 0 for x ≥ 1, f ′ (x) ‹ 0 for -1 ‹ x < 0,

f ′ (x) = 0 for x ≤ -1

What the symbols are telling us:

f(0)=0:   this gives us a fixed point on the sketch, namely  (0,0).  Plot this point first.

f ′ (x) › 0 for 0 ‹ x ‹ 1:   a positive derivative means that the curve is rising from     

                                   left to right. (a positive derivative indicates an increasing

                                   function). So you must sketch the graph rising in this  

                                   interval. Since the graph is at (0,0) at the left end point,

                                   sketch it so it rises from that point up to x=1. We do not

                                   know how it rises, so don’t assume it’s straight.

f ′ (x) = 0 for x ≥ 1:     a zero derivative indicates a “flat point on the curve”. These 

                                 will be places where there are possible max/min values for the

                                 curve. i.e., high or low points.  Since the derivative is always

                                 zero from x=1 and beyond, the curve is flat (constant level)

                                 for values of x ≥ 1. This section will look like a horizontal line.

f ′ (x) ‹ 0 for -1 ‹ x <0:  a negative derivative means that the curve is falling from left

                                  to right. (a negative derivative indicates a decreasing

                                  function). So you must sketch the graph falling in this

                                  interval. So the curve must fall to the fixed point (0,0). We

                                 don’t know how it falls, so don’t assume it is straight.

f ′ (x) = 0 for x ≤ -1:   again, this is similar to one of the above, but for values of x

                                from -1 and to the left. A zero derivative is a “flat

                                point” (could be a possible high or low point, if it occurs at a

                                single value). Since the derivative is zero for all values from -1

                                & to the left, the curve will be at a constant level again there.

Note:  There could be many possible sketches, since we do not know how high the curve is at x=1 (no information given). Any sketch satisfying the conditions given will be accepted. It’s best to use a pencil with an eraser, since students often need to adjust their sketches based on the information given.  See the possible sketch on the solution sheet given in class.