Local Linearity and Estimating the Derivative Graphically
If you can zoom in on f(x) to produce a straight line portion of f(x) containing f(a) and points on either side of
x = a, then you can use the straight line portion to estimate f’(a). This means f’(a) exists. If you cannot produce
the straight line portion, then f’(a) does not exist.
Zoom in on the graph of each function at the point x = 1 to estimate its derivative at the given x value
graphically. Indicate the two points you are using to find the slope at that point. Compare your slope with that
given by the tangent line at the given x value.
1.
f ( x) ! 3 x at x = 1
2.
f ( x ) ! arccos x at x = 1
3.
f ( x) ! x
4.
f ( x) ! cos 2 x at x =
4
3
at x = 0
"
2
Zoom in on the graph of the indicated function to estimate its derivative graphically at x !
"
4
. Compare your
slope with that given by the tangent line at that point.
# $
5.
f ( x) ! sin x 4
6.
f ( x ) ! cos x
Graphically estimate the indicated derivative (if it exists). Compare your slope with that given by the tangent
line at the given point.
7.
f ' (1) : f ( x) ! xe x
8.
f ' (3) : f ( x ) ! x 3 % 27
9.
f ' (1) : f ( x ) ! ln 2 x
10.
f ' (" ) : f ( x) ! sin x
Adapted from James Rahn
8/04