C Roettger, S15 – Math 165, Project 1
The table below gives for each project the function f (x), the x-value b
that is common for all average rates of change in part a), and the value of
m, the slope of the tangent line at b, rounded to six digits after the decimal
point. In chapter 3, we will learn an easier way to compute the exact value
of m, without using limits.
function f (x) x − value b
√
4 − x2
1
4−x
1
1 − cos x
sec x =
m
1.9
−3.042435
3.4
2.777778
6.5 −392.454984
1
cos x
1
−3
1
√
4−x
p
√
4− x
x2
1.2
7.098376
1.5
−5.333333
3.5
1.414214
14.0
−0.131455
x3 + x
2.4 −55.700000
x−2
x+2
1.6 −34.948980
x2 − 2
x−1
1.1
−0.317101
x4 − 1
Here, I will only show a model solution for the first function / value combination, since the others would be very similar in principle, and the choices
of values for a were up to you.
C Roettger, S15 – Math 165, Project 1A
This is about the function
f (x) =
√
4 − x2 .
a) Make a table of the average rate of change of f (x) between x = a and
x = 1.9 for 8 different values of a, approaching 1.9. Compare Example 2,
table 2.1, and problem 19 of 2.1.
b) According to your table, what is your best estimate for the instantaneous
rate of change of f (x) at x = 1.9?
c) Also only by looking at your table from part a), estimate the accuracy
of your estimate in part b). Try to choose your values of a so that you get
roughly 4 - 6 digits of accuracy for the closest value of a.
d) The tangent line to the graph of f (x) at x = 1.9 has equation y =
m(x − 1.9) + f (1.9). Write the equation for the tangent line with your
approximation of the slope m.
Solution a) Here are some values of a that seem relevant to me, and the
average rate of change of f (x) between a and b = 1.9.
a
1.5
1.6
1.7
1.8
1.85
1.89
1.899
1.8999
1.89999
1.899999
1.900001
1.90001
1.9001
fa,b
−1.7459396
−1.9183340
−2.1453279
−2.4727999
−2.7086882
−2.9640858
−3.0342630
−3.0416142
−3.0423528
-3.0424267
-3.0424431
−3.0425170
−3.0432565
b) The two values of a closest to b give rates of change which agree up to the
fourth digit after the decimal point, −3.0424 (see the boxed-in numbers). So
we take this as our best estimate.
c) Again from the table, we would guess that the true value lies between the
numbers we got for a = 1.899999 and a = 1.90001 (boxed). So our error of
our estimate should not be greater than 0.00005.
d) We just substitute our estimate for m and the value of f (b), rounded to
four digits as well (greater accuracy for f (b) would not make sense because
of the error in m)
y = −3.0424(x − 1.9) + 0.6245
This would give a very good approximation of f (x) for values of x close to
b = 1.9.
Remark: a good idea would also have been to take the average of the two
boxed-in numbers above. This would have given an estimate
m ≈ −3.0424349224 . . .
and comparing to the exact value
m = −3.0424349222 . . .
this obviously is much more accurate for little extra effort.
Graphs of f (x) (black) with two secant lines (blue) and the tangent line at
x = 1.9 (red). For the second plot, we zoom in on the point x = 1.9. You
can see how the tangent line is much closer to the graph at this microscopic
scale, so it would give a much better approximation of f (x) for values of x
extremely close to 1.9.