QOD:

Each one of these represents position, velocity and acceleration.

Acceleration is the derivative of velocity, and velocity is the derivative of position.

We will therefore note acceleration as the second derivative, velocity as the derivative and position as the function.

Key point on the graph:

We not that the slope of the green line approaches it's lowest quantity here, and that the y value of the red one reaches it's lowest value. Following that, the blue line reaches a y value of 0 when the red line's slope approaches zero.

Taking this point as representative of the function and its derivatives as a whole, then we assume that the green line is the function, the red line is the derivative, and the blue line is the second derivative.

They match up at all points. At the beginning, green (position)'s slope is zero, and red (velocity) is at zero, while it's slope is at it's lowest value -- and the green point is at it's lowest y value.

We then went over the function:

c(x) = 5000 + 10x + .05x^2

Where c is cost and x is designer glasses.

Asking for a) The average cost of designer glasses of 100 pairs, b) The marginal cost of 100 and c) The change in cost of 101 and 100.

a) asks for the plugging in of 100 as x and then dividing it by 100, finding the cost of producing each pair, at a production level of 100. Doing so:

(5000 + 10(100) + .05(100)^2)/100 = 65

It costs 65 to produce each pair of glasses at 100 glasses being produced.

b) asks for the change in cost at 100. We find this deriving. I used the power rule.

c(x) = 5000 + 10x + .05x^2

c'(x) = 5000(0)(x^-1) + 10(1)(x^0) + .05(2)(x^1)

c'(x) = 10 + .1x

We then substitute x for 100 to find the slope at that particular point.

c'(100) = 10 + .1(100)

c'(100) = 20

So the cost is increasing by 20 at that particular point.

c) asks for the difference between the cost at 101 and 100.

c(101) - c(100)

5000 + 10(101) + .05(101)^2 - 5000 - 10(100) - .05(100)^2

20.05