Can someone please help with these two calculus questions? The velocity function
ID: 2868374 • Letter: C
Question
Can someone please help with these two calculus questions?
The velocity function of an object is given below. Velocity is measured in feet per second and time is measured in seconds. (a) Find the exact distance traveled by the object in the first 60 seconds. feet (b) Find the exact distance traveled by the object in the first 80 seconds. You should sketch the graph. feet (c) What does integrate 60 80 v(t) dt represent? (select all that apply). Area under the graph of v(t) over the interval [60, 80]. Change in position of the object between 60 and 80 seconds. Distance traveled by the object between 60 and 80 seconds. Use the information in the table below to evaluate each of the expressions. If an estimate is needed, use the best Left Hand sum for the interval. Pay attention to the notation.Explanation / Answer
a) First 60 secs :
v(t) = 10
Integrating :
s(t) = 10t (from 0 to 60)
s(t) = 10(60) - 10(0)
s(t) = 600 feet ---> ANSWER
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b) 80 seconds :
Since we are finding distance, we need to take the modulus
Over the first 60 seconds, a total of 600 feet was covered
Now, over t = 60 to t = 80 :
(integral 60 to 80) [(-1/2)(t - 60) + 10] * dt
(integral 60 to 80) [-t/2 + 40] * dt
(-t^2/4 + 40t) (60 to 80)
40(80 - 60) - (1/4)(80^2 - 60^2)
100
So, total distance = 600 + 100 = 700 feet ----> ANSWER
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c)
The integral is nothing but the area
So, it is the area
Since integrating velocity gives us position, a definite integral of velocity will gve us change of position
So, this is also true
Integral of velocity is the displacement, not the distance.
But in this case, over t = 60 to t = 80, distance was positive. So, this is also true
So, check all of them ----> ANSWER
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a)
(int from 0.1 to 0.5) g'(t) * dt
When we integrate g'(t), w eget g(t)
And when we apply limits, we get : g(0.5) - g(0.1)
0.8 - 2.9
-2.1 ----> ANSWER
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b) (int from 0 to 0.3) f(t)*dt
for this, we can use the left end rule
Since we are integrating over 0 to 0.3, the left endpoints are : 0 , 0.1 and 0.2
f(0) = 0.3
f(0.1) = 0.2
f(0.2) = 0.2
Adding : 0.3 + 0.2 + 0.2 = 0.7
Now, multiply this by delta(x), which is 0.1 here
0.7 * 0.1
0.07 ----> ANSWER