How would you do the following derivative problem?

Question

Does anyone know how to do these problems? If you could show how to do it and what the answers would be, it would be greatly appreciated!! Thanks!

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f(p).
A) What is the meaning of the derivative f '(8)? What are its units? B) Is f'(8) positive or negative? Explain.

Answer 1

Generally, the derivative (with respect to a certain variable, say price, or x) indicates the slope of a line that is tangent to the curve at a certain point, here 8. Imagine that you were to graph the function, f(p), and it would show up as a downward-sloping curve (I assume it slopes down because the more expensive the item, the less is demanded, hence the fewer items are sold). That slope might be something like -7dy/dx, which would tell us that as x (here price) goes up one unit, y, here Quantity, goes down 7. The "d" here means the change in, and the derivative is equivalent to dQ/dP. The units depend on how you measure the values.

I cannot determine the actual value of the derivative unless I know the function that shows how Q is related to p (f(p)). I suspect the derivative is negative generally, but it may not have the same value (unlike a straight line, which has a constant slpe, like -7, many functions have slopes of different values, depending upon the value of x. Therefore it is possible that the slope changes value at various levels of x, so this is why a derivative should be calculated at a particular point, like 8.

There are some general and pretty easy rules for calculating a derivative. An example is y=4x. Here the derivative is simply 4. But for y=4x^2, the rule is we multiply the power to which x is raised by x, so here we have 2*4 and then lower the power by 1, giving 8x as the derivative (actualy the first derivative) of the function with respect to x. We can take a second derivative of 8x, giving 8. The second derivative of a function shows the change not in the slope, but in the derivative. A function may have a positive first derivative (indicating that y grows as x grows) but a negative second derivative, meaning that the derivative (change in Y per change in x) becomes smaller. In economics, for example, we can imagine that as people's incomes grow, they spend more (so we model spending versus income, of S=f(I) where S is spending and I is income). We expect a positive first derivative since spending will go up as income does. But people don't spend more up to infinity, because at some point they will have purchased most items they need and the rest they save. Therefore, as income goes higher and higher, the change in spending is lower and lower. This is one reason why money that is redistributed to the poor is almost always spent while money redistributed to therich is often saved- the rich have a lower propensity to spend.

Answer 2

The first derivative of a function is the rate of change of that function. f ' (8) is the rate of change of the function f at p = 8.

Be careful here, as this is something of a trick question. The normal way of expressing the selling of coffee would be to have pounds as the independent variable and price in dollars depending on how many pounds are purchased. Here it is reversed, so Q (p) represents how much coffee would be sold at different prices. If they ask too much for gourmet coffee, the amount sold may drop off dramatically. My guess would be that the graph looks like a log graph. If the coffee was $ 0.50 per pound for something like Starbucks, the amount would be very high; when it gets to $25 per pound the quantity would be very low.

But in general: the derivative of a function is the rate of change, or slope, of that function at any given point on the x axis (which is the p axis in this example). So the first derivative would, in this case, probably be a straight line tangent to that log graph at any given point, but would be different at every point approaching a limit. After all, they would probably sell 0 pounds at $200 per pound, and the same amount at $10,000 per pound. But if you were to pick a point p, say p = 8, the graph of f ' would be a straight line tangent to f at p = (8).

Please keep in mind that the above explanation made a lot of assumptions about the behavior of the quantity of coffee sold at varying prices.

Answer 3

This one is a very simple series of power laws. L'(x)=d/dx[20*x]-d/dx[4*x^2]-d/dx[3/2*... L'(x)=20-8*x-3*pi*x L'(x)=20-(8+3*pi)*x

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