Question

Show that the average rate of change of f(x) = x^3 over [1,x] is equal to x^2 + x +1. (I managed to answer this.)

Use this estimate the instantaneous rate of change of f(x) at x=1. (I need help answering this, please.)

Answer 1

If you're estimating the instantaneous rate if change at a point but you only have an average rate of change formula, the best way to get an accurate estimate is to simply minimize the average as much as possible. So you can plug in x=1.001 or x=1.0001 and as you get closer and closer to one (infinitely close even), and the difference between 1 and x gets smaller, you get closer and closer to the instantaneous rate of change.

Answer 2

do i plug in numbers < 1 if the domain is [1,x]?

Answer 3

you can but you're trying to find the instantaneous rate of change at 1, so you want your average rate of change to be as close to that as possible, and that is means making the average between your x and 1 as close to 1 as possible. Since we have a formula for the average rate of change, if our x IS one, then we should get the smallest possible (trivial) distance between x and 1, so plug in 1 to get the instantaneous rate of change at one.

Answer 4

the problem wants me to estimate the instantaneous rate of change

Answer 5

i know how to find the exact value, but not how to estimate - i think i should plug in the 1.01, etc

Answer 6

Your formula is the average rate of change over a range, and as you make that range smaller and smaller, the formula converges on the result. If you only want an estimate then just make that range small, but not zero.

Answer 7

i drew a table with f(x) as the top row and 'x' as the bottom row

Answer 8

what do you mean by the formula converging on the result?

Answer 9

The formula gets closer and closer to the actual value but only reaches it when your interval between 1 and x is zero. This means the interval between 1 and x is infinitely small. The formula will eventually reach the right answer if you use enough decimal places

Answer 10

i see. i'm not allowed to use a calculator on this problem. is there an alt way to solve this?

Answer 11

Just plug in 1. It gives the exact same answer were you using calculus.

Answer 12

i wish it were as simple as that, but the problem wants an estimate :( i think i need to use average velocities/velocity, but i don't know how i would

Answer 13

Well since you can't be sure the answer is the instantaneous rate of change, you are estimating.

Answer 14

howcome i can't be sure the answer is/isn't that? curious

Answer 15

Have you been taught how to calculate instantaneous rate of change yet?

Answer 16

my teacher taught me that, but used an example different from this one. and she never taught me how to estimate the instant. v

Answer 17

'v' meaning rate of change

Answer 18

Well an estimate is a guess. You have a formula that allows you to calculate the average rate of change between 1 and any other number. But what if that other number IS 1, then that would give you the average rate of change between 1 and 1. If you are super concerned about it being an 'estimate' then just use values close to, but not exactly 1.

Answer 19

Here is the way to do it
\[\frac{f(x)-f(1)}{x-1}=\frac{x^
3-1}{x-1}=\frac{(x-1)
\left(x^2+x+1\right)}{x-1}=
x^2+x+1

\]
if\( x\) approaches 1, then the ratio approaches 3