Write and simplify the difference quotient for the function b(t)= 3t^2-4t+5 over the interval [7,7+h]

b(t+h)-b(t)

=(3(t+h)^2-4(t+h)+5) - (3t^2-4t+5)
=(3t²+6ht+3h² -4t-4h + 5) - (3t^2-4t+5)
=6ht+3h²
Substitute t=7 to get an expression in terms of h.

To find the difference quotient for the function b(t) = 3t^2 - 4t + 5 over the interval [7,7+h], we need to determine the slope of the secant line passing through the points (7, b(7)) and (7+h, b(7+h)).

First, we find the values of b(7) and b(7+h):

b(7) = 3(7)^2 - 4(7) + 5
= 147 - 28 + 5
= 124

b(7+h) = 3(7+h)^2 - 4(7+h) + 5
= 3(49 + 14h + h^2) - 28 - 4h + 5
= 147 + 42h + 3h^2 - 28 - 4h + 5
= 3h^2 + 38h + 124

Now, we can write the difference quotient as follows:

[b(7+h) - b(7)] / [(7+h) - 7]
= [3h^2 + 38h + 124 - 124] / h
= (3h^2 + 38h) / h
= 3h + 38

Therefore, the simplified difference quotient for the function b(t) = 3t^2 - 4t + 5 over the interval [7,7+h] is 3h + 38.

To find the difference quotient for a function, we need to evaluate the function at two different points within the given interval and divide the difference by the length of the interval.

First, let's evaluate the function b(t) at the two points: t = 7 and t = 7 + h.

b(7) = 3(7)^2 - 4(7) + 5
= 147 - 28 + 5
= 124

b(7 + h) = 3(7 + h)^2 - 4(7 + h) + 5
= 3(49 + 14h + h^2) - 28 - 4h + 5
= 147 + 42h + 3h^2 - 28 - 4h + 5
= 124 + 38h + 3h^2

Now, we can write the difference quotient:

[b(7 + h) - b(7)] / h

Substituting the evaluated values:

[(124 + 38h + 3h^2) - 124] / h

Simplifying by canceling out the terms:

[38h + 3h^2] / h

Now, we can simplify it further:

38 + 3h

So, the simplified difference quotient for the given function b(t) = 3t^2 - 4t + 5 over the interval [7, 7 + h] is 38 + 3h.