Velocity and other rates of change

[noahstaz]

Hi I need to solve this question using limits and I have no idea how to #8.

Thanks!!

 

[skeeter]

the question wants you to determine [MATH]V'(10)[/MATH]
you are given [MATH]V(t) = 1000\left(1 - \dfrac{t}{60}\right)^2[/MATH]
are you capable of calculating [MATH]V'(t)[/MATH] using the chain rule and then substituting 10 for t to determine the instantaneous rate of change?

 

[noahstaz]

the question wants you to determine [MATH]V'(10)[/MATH]
you are given [MATH]V(t) = 1000\left(1 - \dfrac{t}{60}\right)^2[/MATH]
are you capable of calculating [MATH]V'(t)[/MATH] using the chain rule and then substituting 10 for t to determine the instantaneous rate of change?

hi I am supposed to solve using one of the tso formulas here

 

[skeeter]

good for you! I recommend this form ...

[MATH]V'(10) = \lim_{t \to 10} \dfrac{V(t) - V(10)}{t - 10}[/MATH]
... go for it!

 

[skeeter]

[MATH]\lim_{t \to10} \dfrac{1000\left(1-\frac{t}{60}\right)^2 - 1000\left(1 - \frac{10}{60}\right)^2}{t-10}[/MATH]
[MATH]1000 \cdot \lim_{t \to 10} \dfrac{\left(1 - \frac{t}{30} + \frac{t^2}{60^2}\right)-\left(\frac{5}{6}\right)^2}{t-10}[/MATH]
[MATH]1000 \cdot \lim_{t \to 10} \dfrac{\frac{t^2}{3600} -\frac{t}{30} + \frac{11}{36}}{t-10}[/MATH]
[MATH]\dfrac{1000}{3600} \cdot \lim_{t \to 10} \dfrac{t^2-120t+1100}{t-10}[/MATH]
ok ... most of the grunt work algebra is done. You finish it up.

 

[HallsofIvy]

Note that if that is going to have an answer then the numerator will have to have a factor of t- 10 in order to cancel the t- 10 in the denominator.

Knowing that t- 10 is one factor if \(\displaystyle t^2- 120t+ 1100\) what is the other?