Recall that the average velocity between times t1and t2 is calculated using

vave = y(t2) − y(t1)/t2 − t1.

We are given that
y = 14t − 1.86t^2.
Evaluating at
t = 1,
we have
y(1) = 14(1) − 1.86(1)2 = 12.14.
Thus, the average velocity between times 1 and 1 + h is as follows.
vave = y(1 + h) − y(1)/(1 + h) − 1

= [14(1 + h) − 1.86(1 + h)^2] − 12.14/h
=10.28h − _____h^2/h

= ____ − _______h, h ≠ 0

To find the average velocity between times 1 and 1 + h using the given equation y = 14t − 1.86t^2, we can substitute the values into the equation for vave:

vave = [14(1 + h) - 1.86(1 + h)^2] - 12.14/h

Expanding and simplifying the equation gives:

vave = 14 + 14h - 1.86(1 + 2h + h^2) - 12.14/h

= 14 + 14h - 1.86 - 3.72h - 1.86h^2 - 12.14/h

= -1.86h^2 + (14 - 3.72 - 12.14/h)h + 14 - 1.86

= -1.86h^2 + (0.14 - 12.14/h)h + 12.14

= -1.86h^2 + 0.14h - 12.14 + 12.14/h

The expression for average velocity between times 1 and 1 + h is -1.86h^2 + 0.14h - 12.14 + 12.14/h, where h ≠ 0.

To find the average velocity between times t1 and t2, we need to substitute the values of y(t2) and y(t1) into the formula vave = (y(t2) - y(t1)) / (t2 - t1).

Given that y = 14t - 1.86t^2, we can evaluate y(1) by substituting t = 1 into the equation:

y(1) = 14(1) - 1.86(1)^2
= 14 - 1.86
= 12.14

So y(1) is 12.14.

Now, we want to find the average velocity between times 1 and 1 + h. Substituting t = 1 + h and t = 1 into the equation, we have:

vave = [14(1 + h) - 1.86(1 + h)^2] - 12.14 / (1 + h) - 1
= 14(1 + h) - 1.86(1 + h)^2 - 12.14 / h

Expanding (1 + h)^2, we get:

vave = [14(1 + h) - 1.86(1 + 2h + h^2)] - 12.14 / h
= 14 + 14h - 1.86 - 3.72h - 1.86h^2 - 12.14 / h
= 10.28h - 1.86h^2 - 12.14 / h

Simplifying further, we have:

vave = 10.28h - 1.86h^2 - 12.14 / h

Now, this expression is not yet simplified completely. To simplify it, we need to factor out h from the numerator:

vave = (10.28h - 12.14) / h

Therefore, the average velocity between times 1 and 1 + h is (10.28h - 12.14) / h, where h is not equal to 0.

You have trouble plugging the values into your expressions?

(-1.86h^2 + 10.28h + 12.14 - 12.14)/h
-1.86h + 10.28