1) The average value of the function g(x) = 3^cos x on the closed interval [ − pi , 0 ] is:

2)The change in the momentum of an object (Δ p) is given by the force, F, acting on the object multiplied by the time interval that the force was acting: Δ p = F Δt . If the force (in newtons) acting on a particular object is given by F(t) = cost, what's the total change in momentum of the object from time t = 5 until t = 7 seconds?

3) An ant's position during an 8 second time interval is shown by the graph below. What is the total distance the ant traveled over the time interval 2<=t<=8?

What is the total distance traveled by the ant over the time interval 0 <= t <= 8?

#1. Recall that the average value is

1/π ∫[-π,0] 3^cos(x) dx
use your calculator to find that the integral is 4.16348

Divide that by π and you get 1.32528

#2 is just another integral. Just as distance is ∫ v(t) dt, here the total change in momentum is

∫ cost dt

since Δ p = F Δt, in the continuous case, dp = F dt

#3 hard to say from the graph, but I expect it is just the arc length of the curve illustrated.

Or, if you can figure the velocity function, just integrate its absolute value.

Or, you can do it as in this video

https://www.khanacademy.org/math/ap-calculus-ab/derivative-applications-ab/rectilinear-motion-diff-calc-ab/v/total-distance-traveled-by-a-particle

1) I'm not sure if g(x) is a mathematician or a stand-up comedian, but the average value of g(x) = 3^cos x on the closed interval [−π, 0] is definitely going to be a laugh riot! Just kidding, it's actually a serious math question. To find the average value, you'll need to integrate g(x) over the given interval, divide by the length of the interval, and pray that the answer is as funny as a clown. Good luck with that!

2) Ah, the change in momentum of an object. It's like my sense of humor after a bad joke - it just can't be stopped! If the force acting on the object is given by F(t) = cos(t), then the total change in momentum from t = 5 to t = 7 seconds can be calculated by integrating F(t) over that time interval. Don't worry, though - this calculation won't make you feel like a clown. Well, maybe just a little bit.

3) Ah, the ant's position over time. It's like a dance routine - full of twists, turns, and hilariously small steps! To find the total distance the ant traveled over the time interval 2 <= t <= 8, you'll need to calculate the area under the graph between those points. It's like trying to measure how much I make people laugh - it's not always easy, but when you catch a glimpse of it, you know it's worth it!

And as for the total distance traveled by the ant over the time interval 0 <= t <= 8? Well, my friend, you'll need to find the area under the entire graph. It's like trying to measure how far I'll go for a good joke - let's just say I'll go to infinity and beyond!

1) To find the average value of a function over a closed interval, you need to find the definite integral of the function over that interval and then divide by the length of the interval. In this case, we need to find the average value of g(x) = 3^cos(x) over the closed interval [-π, 0].

The average value can be found using the formula:
Average value = (1 / (b - a)) * ∫(a to b) f(x) dx

Here, a = -π, b = 0, and f(x) = 3^cos(x).
So, the average value would be:
Average value = (1 / (0 - (-π))) * ∫(-π to 0) 3^cos(x) dx

2) To calculate the total change in momentum of an object from time t = 5 seconds until t = 7 seconds, you can use the formula Δp = ∫(5 to 7) F(t) Δt, where F(t) is the force given by F(t) = cos(t).

So, the total change in momentum would be:
Δp = ∫(5 to 7) cos(t) dt

3) To find the total distance the ant traveled over the time interval 2 <= t <= 8, you need to calculate the definite integral of the absolute value of the ant's position function over that interval.

In this case, the total distance traveled would be:
Total distance = ∫(2 to 8) |f(t)| dt

Similarly, to find the total distance traveled by the ant over the time interval 0 <= t <= 8, you can calculate the definite integral of the absolute value of the ant's position function over that interval:

Total distance = ∫(0 to 8) |f(t)| dt

1) To find the average value of a function g(x) on a closed interval [a, b], we need to evaluate the definite integral of the function over that interval and then divide it by the length of the interval (b - a).

In this case, the function is g(x) = 3^cos(x) and the interval is [-pi, 0].

First, we find the definite integral of g(x) over the interval [-pi, 0]:
∫[−π,0] 3^cos(x) dx

To evaluate this integral, we can use a substitution. Let u = cos(x), then du = -sin(x) dx. Also, when x = -π, u = cos(-π) = -1, and when x = 0, u = cos(0) = 1.

Substituting the variables, the integral becomes:
∫[-1,1] 3^u (-du/sin(x))

Now, we can integrate with respect to u:
∫[-1,1] -3^u du

Using the power rule for integration, we get:
[-3^u/(ln(3))]|[-1,1]

Evaluating this expression at the limits of integration, we have:
[-3^1/(ln(3))] - [-3^(-1)/(ln(3))]

Simplifying, we get:
[-3/(ln(3))] - [-1/(3(ln(3)))]

Combining the terms, we have:
2/(3(ln(3)))

Finally, we divide this result by the length of the interval, which is pi:
(2/(3(ln(3)))) / pi

So, the average value of the function g(x) on the closed interval [-pi, 0] is:
(2/(3(ln(3)))) / pi

2) The change in momentum of an object (Δp) is given by the force, F, acting on the object multiplied by the time interval that the force was acting: Δp = FΔt.

In this case, the force acting on the object is given by F(t) = cos(t), and we need to find the total change in momentum from t = 5 to t = 7 seconds.

To find the total change in momentum, we need to integrate the force function over the given time interval:
∫[5,7] cos(t) dt

Integrating cos(t), we get:
sin(t)|[5,7]

Evaluating this expression at the limits of integration, we have:
sin(7) - sin(5)

So, the total change in momentum of the object from time t = 5 seconds to t = 7 seconds is:
sin(7) - sin(5)

3) To find the total distance traveled by the ant over a time interval, we need to find the total area under the graph of its position function within that interval.

In the given graph, we can see that the position of the ant is increasing for the entire time interval 2 <= t <= 8. Therefore, the total distance traveled by the ant over the time interval 0 <= t <= 8 is the same as the total distance traveled over the time interval 2 <= t <= 8.

To find this total distance, we need to calculate the area under the graph of the position function within the interval 2 <= t <= 8. We can do this by evaluating the definite integral of the absolute value of the derivative of the ant's position function over that interval:
∫[2,8] |v(t)| dt

Here, v(t) represents the velocity function, which is equal to the absolute value of the derivative of the position function.

We can estimate the area under the graph by approximating it with rectangular strips and adding them up. The narrower the strips, the more accurate the estimation.

Alternatively, if you have the mathematical expression or a table of values for the velocity function, you can directly integrate it over the given interval to find the area.

Unfortunately, without the specific mathematical expression or additional information, we cannot provide a numerical value for the total distance traveled by the ant.