Written Homework 7 Solutions

11. (a) Sketch the graph of the function y = 2x + 5
on the interval 0.2 ≤ x ≤ 4.
x
Solution:
20
15
10
5
0 0.5 1 1.5 2 2.5 3 3.5
(b) Where is the lowest point on that graph? Give the value of the x-coordinate exactly.
Solution: The lowest point on this graph occurs when 0.2 ≤ x ≤ 4 and f ′ (x) = 0.
Since f(x) = 2x + 5
x , f ′ (x) = 2 − 5
, and after some simple algebra, we see that
x2 f ′
5
(x) = 0 when x = . Thus the lowest point on this graph is ( , 6.3246).
5
2
13. This problem was not graded and is very similar to 15.
15. (a) Write the microscope equation for y = √ x at x = 3600.
Solution: The microscope equation is y ≈ √ 3600 + △y = 60 − 1
· △x.
120
(b) Use the microscope equation to estimate √ 3628 and √ 3592. How far are these estimates
from the values given by a calculator?
Solution: Using the above microscope equation, we will first estimate √ 3628. To
figure out our estimate for y, we first need to find △x: △x = 3600 − 3628 = −28.
Now we can plug into y ≈ 60 − 1
1
· △x. So y ≈ 60 − · (−28) ≈ 60.23333. If we
120
plug √ 3628 into a calculator we get 60.23388, so there is very little discrepancy.
Now we can estimate √ 3592. We first need △x: △x = 3600 − 3592 = 8. Thus
y ≈ 60 − 1
1
· △x. So y ≈ 60 −
120 120 · (8) ≈ 59.93333. If we plug √ 3628 into a
calculator we get 59.933296, so again there is very little discrepancy.
17. A ball is held motionless and then dropped from the top of a 200 foot tall building. After
t seconds have passed, the distance from the ground to the ball is d = f(t) = −16t 2 + 200
feet.
3
120
2