What Are the Solutions to These Calculus Problems?

[Carl_M]

hints? Derivatives: Intervals, stationary points, logarithms, continuous functions

Homework Statement

Got any hints or anything?

1. Suppose that f(x) = (x - 3)^4 ( 2x + 5)^5
a) Find and simplify f ' ( x )
b) Find stationary points of f
c) Find exactly the intervals where f is increasing and intervals where f is decreasing

2. Find the stationary points of g(x) = 2cos - sqrt(3)x , 0< _ x < _ 2pi and classify them (as local minimum, local maximum or neither).

3. The temperature of the ground at a distance of d centimetres below the surface at a certain location can be modeled by g(t) = 16t + 11e^-0.00706dCOS(2(pi)(t) - 0.00706d-0.628)
where t is the time in years since July 1.
a) Find and interpret g(t) and g '(t) on sept 1 at ground level (d =0)
b) Find and interpret g(t) and g '(t) on sept 1 at 3 m below ground level.

4. Let h be continuous, differentiable function such that g(3) = -7, g(-7) = 3, g '(3) = 2, and g '(-7) = 4

a) Find (g^-1)(3) and (g^-1) '(3)
b) Find an equation for the tangent line to the graph of g^-1(x) at x=3
c) With only the information, what is your best estimate of (g^-1)(4) ?

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi Carl_M!

(have a pi: π and a square-root: √ and a ≤ and try using the X² tag just above the Reply box )

Show us what you've tried, and where you're stuck, and then we'll know how to help!

Start with 1. ​

 

[Carl_M]

Hi Carl_M!

(have a pi: π and a square-root: √ and a ≤ and try using the X² tag just above the Reply box )

Show us what you've tried, and where you're stuck, and then we'll know how to help!

Start with 1. ​

1. a) d/dx (( x-3)^4(2x+5)^5)

= (2x+5)^5(d/dx((x-3)^4)) + (x-3)^4(d/dx((2x+5)^5)
=4(x-3)^3(2x+5)^5(d/dx(x-3)) +5(x-3)^4(2x+5)^4(d/dx(2x+5))
=4(x-3)^3(2x+5)^5 +10(x-3)^4(2x+4)^4 = 0

X=3 , X = 5/9 , -5/2

How would I get the increasing/decreasing?

c ) Would it be Increase on [ -oo, 5/9] and decrease on [5/9, oo] ?

 

[tiny-tim]

Sorry, that's too difficult to read …

please use the X² tag just above the Reply box ​