To sketch: The graph of the function f(t)=t2−5t+4 on the interval [0,6].
b.
To determine
To compute: The area function A(x) and draw its graph.
c.
To determine
To show: The local extrema of A occur at the zeros of f.
d.
To determine
To explain: The geometrical and analytical explanation for observation in part (c).
e.
To determine
To find: The approximate zeros of A, other than 0, and call them x1 and x2, where x1<x2.
f.
To determine
To find: The value of b such that the area bounded by the graph of f and the t-axis on the interval [0,x1] equal to the area bounded by the graph of f and the t-axis on the interval [x1,b].
g.
To determine
Whether the local extrema of A occurs are zeros of f or not.
Find the area of the shaded region.
(a)
5-
y
3
2-
(1,4)
(5,0)
1
3
4
5
6
(b)
3 y
2
Decide whether the problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to
estimate the solution.
STEP 1: Consider the figure in part (a). Since this region is simply a triangle, you may use precalculus methods to solve this part of the problem. First determine the height of the triangle and the length of the triangle's base.
height 4
units
units
base
5
STEP 2: Compute the area of the triangle by employing a formula from precalculus, thus finding the area of the shaded region in part (a).
10
square units
STEP 3: Consider the figure in part (b). Since this region is defined by a complicated curve, the problem seems to require calculus. Find an approximation of the shaded region by using a graphical approach. (Hint: Treat the shaded regi
as…
Solve this differential equation:
dy
0.05y(900 - y)
dt
y(0) = 2
y(t) =
Suppose that you are holding your toy submarine under the water. You release it and it begins to ascend. The
graph models the depth of the submarine as a function of time.
What is the domain and range of the function in the graph?
1-
t (time)
1 2
4/5 6 7
8
-2
-3
456700
-4
-5
-6
-7
d (depth)
-8
D: 00 t≤
R:
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