rate of change, horizontal tangent, and derivative

[freeshine]

it's online homework so i can tell right away what is right/wrong.
1. if h(5) = 3 and h'(5) = -5, find d/dx (h(x)/x) at x=5

i dont know where to start with this one. i thought the answer would be 5,3,-5, -1,1, etc. but i tried all of those and they were all wrong.

2. in this exercise we estimate the rate at which the total personal income is rising. in 1999, the population was 961500 and was increasing at 9200 people per year. The average annual income was $30583 per capita and was increasing at $1400 per year. use the product rule and these figures to estimate the rate at which total personal income was rising.

i thought this would be 9200/1400 but that wasnt right. i dont know where the product rule comes into play either.

3. find the points on the curve y = 2x^3 + 3x^2 - 36x + 7 where the tangent is horizontal. enter your answer as an ordered pai in the form (x,y).

for this one, i found the derivative, set it equal to 0 and got (-3,34) and (2,37) but apparently those were wrong.

thanks for any help!

 

[galactus]

3. find the points on the curve y = 2x^3 + 3x^2 - 36x + 7 where the tangent is horizontal. enter your answer as an ordered pai in the form (x,y).

for this one, i found the derivative, set it equal to 0 and got (-3,34) and (2,37) but apparently those were wrong.

thanks for any help!

It appears you just have some arithmetic errors. Run those again because your x values are correct.

I get (2, -37) and (-3,88)

 

[freeshine]

those worked. thanks for your help!

 

[stapel]

1) Use the Quotient Rule, with f(x) = x, so:

. . . . .g(x) = h(x)/f(x)

. . . . .g'(x) = dg/dx = [h'(x)f(x) - h(x)f'(x)] / [f(x)]²

Note that f'(x) = 1 so, in particular f'(5) = 1.

2) Use the Product Rule, with f(x) = (population count) and g(x) = (income per person), so h(x) = f(x)g(x) = (income per person)(people) = (total income).

Eliz.