A
tangent
line to a curve is supposed to be the best straight line approximation
to the curve near the point of tangency. If our curve is the graph of a
function, y = f(x), and the line y = L(x) is tangent to the
curve at the point (a, f(a)), then when b is close to a,
L(b)
ought to be close to f(b).
In
the following exercises, you'll use the formula for a tangent line to the
graph of a function to get approximations to outputs of the function. Of
course, the slope of the tangent line to the graph of y = sin(x)
at x=2.3 is the derivative of the sine function evaluated at x=2.3, and
if our calculator is sufficiently familiar with the sine function to compute
its derivative when x=2.3 (which it is), then it also knows enough to give
us very accurate values for sin(b) when b is close to 2.3
(and it does). So these estimates are only useful when they're all we have;
that is, when we don't have a formula or an accurate calculation procedure
for our original function. So here's some tabular data:
| x | 5.0 | 5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.6 | 5.7 | 5.8 | 5.9 | 6.0 |
| f(x) | 12.12 | 13.32 | 14.28 | 15.04 | 15.98 | 16.66 | 17.34 | 18.10 | 18.92 | 19.36 | 19.98 |
Exercises:
1. Estimate the slope of the tangent line to the graph of f at x=5.2. Use your estimate to find an equation for that tangent line. Use your equation to estimate f(5.22).
2. Repeat exercise #1 at 5.1, using the tangent line at x=5.1 to estimate f(5.09).
3. Continue, using different points, until you get bored.
4. Use a tangent line to estimate the value of the number c for which f(c)=17.00.
5. Suppose you know that G(2.3)=476.92 and that G'(2.3)=-12.4. Find an equation for the line tangent to the graph of G at the point (2.3, 476.92). Use your equation to estimate G(2.286). Use your equation to estimate G(2.307). Use your equation to estimate G(2.285).
6. What would you need to know about the shape of the graph of G to predict whether your estimates from the previous exercise were too big or too small?
7. Use the information about G above to estimate the input to the function G which will produce the output 475. (That is, estimate c such that G(c)=475.00.) Approximately what input will produce the output 480?