Calculus, Im stuck on this.

[jspeicher]

Okay, if T is number of years since 1993, P is the population in billions, with the function:

P=f(t)= 1.15(1.014)^t

Estimate f(6)

Estimate f '(6)

for f(6), i get 1.25 billion

for the derivative at 6, do you guys get 0.0174 billion/yr? I think this is correct but I do not get this answer at all, and I have no clue why I cant.

 

[fatty4ksu]

Yes

 

[dudeman007]

E=MC^2

 

[jspeicher]

my derivative of the exponential function must be way off...

1.15(1.014)^6 * (1.15(1.1014^h)-1) ????

 

[Chaotic42]

The derivative of the function at 6 is:
1.15*(1.014^6)*(ln 1.014)

Edit: This is because the derivative of a^x is (a^x)*(ln a). Try it with a=e.

 

[Soccer55]

The derivative of a^x is (a^x)*ln(a). Use that formula to get your derivative and plug in 6 for x.

-Tom

 

[DrPizza]

For what it's worth, if you forget the rule for the derivative of a^x,
you can ln both sides:
y = a^x
lny = ln a^x
which equals
lny = x*lna
then take the derivative implicitly
1/y dy/dx = lna
solving for dy/dx
dy/dx = lna * y
substituting for y
dy/dx = lna * a^x
or dy/dx = a^x * lna

Wow, so neat to see where everyone else is at in their calculus classes and the level of difficulty their problems are at. I *just* did that lesson on Friday, along with derivatives of logarithmic functions and derivatives of inverse trig functions.
M,T,and W of this week are nothing more than practice/homework days based on that lesson, with problems rapidly increasing in level of difficulty.

 

[ProviaFan]

Interesting... I just "re-learned" that a few weeks ago, having learned (and subsequently forgotten) it previously in high school. 😱

 

[fatty4ksu]

natural log is your friend.....seriously.