Question

1.A natural law of growth is of the form y=
`5e^(0.2x)`. Draw a graph for this law for values of x from x = -3 to x = 3. From the graph:

A)Find the gradient of the curve at x = -2 and x = 1 by differentiation and by drawing a tangent to the curve.
B)Compare your two solution for these gradients

Answer 1

A) The derivative of the function is
.. y' = e^(0.2x)
so the gradients are
.. at x=-2, e^-0.4 ≈ 0.67
.. at x=1, e^0.2 ≈ 1.22
It is difficult to get a good estimate of the gradient by drawing a tangent line. (If you want to try, the second picture is the same graph with the tangent lines turned off. You can print it out and draw your tangent line--or use your favorite picture markup program.)

B) They will likely be within "experimental error", accurate to within a couple of degrees of rotation. (Higher slopes are more difficult to approximate accurately.)