A particle moves at varying velocity along a line and s= f(t) represents the particle's distance from a point as a function of time, t. sketch a graph for f if the average velocity of the particle between t=2 and t=6 is the same as the instantaneous velocity at t=5.

I know that instantaneous velocity is calculated as s(a+h)-s(a)/h but i don't understand what to put in for h. I'm stuck at this point and would appreciate any guidance on what to do next. thanks so much!!!

" I know that instantaneous velocity is calculated as s(a+h)-s(a)/h ...."

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Well sort of.
The limit of that expression as s goes to zero is the instantaneous velocity. It is also the slope of your curve at t = 5.
However the expression as you wrote it is the average velocity or slope of the curve between t = 2 and t = 5
Therefore draw a slope and s value at t = 5
To make it really simple, sketch a straight line with that slope from t = 2 to t = 6
Of course the curve could take all kinds of twists and turns instead of what we sketched, but the basic idea is that at t = 5 the slope is the same as the slope of a straight line from t = 2 to t = 5

To solve this problem, we need to find the function f(t) that satisfies the given conditions.

Let's start by defining the average velocity between t = 2 and t = 6. The average velocity is calculated as the change in position divided by the change in time:

Average velocity = (s(6) - s(2))/(6 - 2)

Next, we need to determine the instantaneous velocity at t = 5. The instantaneous velocity is the derivative of the distance function with respect to time at t = 5:

Instantaneous velocity = f'(5)

To make the average velocity equal to the instantaneous velocity, we equate the two expressions:

(s(6) - s(2))/(6 - 2) = f'(5)

Now, let's integrate both sides of the equation with respect to t to find the function f(t):

∫[(s(6) - s(2))/(6 - 2)] dt = ∫f'(5) dt

[s(t)]|2^6 = f(t) + C

s(6) - s(2) = f(6) + C - (f(2) + C)

s(6) - s(2) = f(6) - f(2)

Rearranging the equation, we have:

f(6) - f(2) = s(6) - s(2)

This means that the change in the function f(t) between t = 2 and t = 6 is equal to the change in position of the particle over the same time interval. Therefore, the graph of f(t) will be a line connecting the points f(2) and f(6) with the same slope as the line connecting s(2) and s(6). The y-intercept of the line will depend on the value of the constant C.

I hope this helps you understand what to do next in solving this problem! Let me know if you have any further questions.

To determine the value of h in the formula for instantaneous velocity, let's take a step back and review the concept of average velocity and instantaneous velocity.

Average velocity is defined as the total displacement divided by the total time taken. In the given problem, we are told that the average velocity between t=2 and t=6 is equal to the instantaneous velocity at t=5.

To find the average velocity, we can use the formula:

Average velocity = (Change in distance)/(Change in time)

Let's consider the time interval from t=2 to t=6. The change in distance during this time can be calculated as f(6) - f(2), where f(6) is the particle's distance from a point at t=6, and f(2) is the particle's distance from the same point at t=2.

Similarly, the change in time is 6 - 2.

Now, we need to calculate the instantaneous velocity at t=5 using the formula you mentioned:

Instantaneous velocity = (f(5+h) - f(5))/h

Here, h represents an infinitely small change in time around t=5. It is a very small value that approaches zero. Think of it as an infinitesimally small time increment.

To satisfy the condition that the average velocity between t=2 and t=6 is the same as the instantaneous velocity at t=5, we need to make these two velocities equal to each other.

So, set up an equation:

(f(6) - f(2))/(6 - 2) = (f(5+h) - f(5))/h

Now, you can solve this equation for h by cross-multiplying and simplifying. This will give you the value of h, which represents the infinitesimally small change in time around t=5.

Once you have determined the value of h, you can substitute it back into the formula for instantaneous velocity to compute the actual value at instantaneous velocity at t=5.

Finally, to sketch the graph of f(t), you can plot different points on the graph using the function f(t). Additionally, you can label the points t=2, t=5, and t=6 on the x-axis to help visualize the given scenarios.