Calculus Problem?

Question

The Olympic flame at the 1992 Summer Olympics was lit by a flaming arrow. As the arrow moved d feet horizontally from the archer, assume that its height h, in feet, was approximated by the function
h = -0.002d2 + 0.7d + 6.6.
Find the relative maximum of the function.

Answer 1

The first thing you need to do is find the derivative.

So:
h'(d) = (2)(-0.002)(d) + 0.7 + 0
h'(d) = -0.004d + 0.7

Now we have to find the critical numbers (points where the derivative is equal to zero). When the derivative is equal to zero, it means the slope of the tangent line equals zero. The point where the tangent line is zero is where the original function (in this case h(d) ) can have a maximum or minimum.

So:
-0.004d + 0.7 = 0
0.7 = 0.004d
0.7 / 0.004 = d
175 = d

Our only critical number is 175. Now we must test it to see that it is in fact a maximum. We do this by picking a number a little less then 175 and a number a little greater then 175. Let's choose 174 and 176. Now plug them into the derivative.

-0.004(174) + 0.7 = 0.004
POSITIVE RESULT

-0.004(176) + 0.7 = -0.634
NEGATIVE RESULT

As we can see, the function goes from positive (function going up) to negative (function going down) at 175.

Now we have to plug it back into the original:

-0.002(175)^2 + 0.7(175) + 6.6
=67.85

Answer: 67.85 feet

Answer 2

You have a parabola; it is the vertex that you want to find.
h(d) = -.002d^2 + 0.7d + 6.6
h'(d) = -0.002(2)d + 0.7 =set= 0
d = -0.7/(-0.004) = 175 the point at which h(d) is a relative max
h(175) = -0.002(175)^2 + 0.7(175) + 6.6
= -.002(30625) + 0.7(175) + 6.6
= -61.25 + 122.5 + 6.6
= 67.85

Answer 3

find the derivative with respect to d and set equal to zero.
dh/dd=0, or graph it since its just an upside down parabola, the peak will be the maximum

Answer 4

max when the derivative, -0.004d + 0.7 = 0, so
d = -0.7/-0.004 = 175,
at which point
h = -0.002(175)² + 0.7(175) + 6.6
h = 67.85 ft.