Differentiation

[as_xoxo]

2. A dinosaur’s height is related to the length of its backbone.In particular, if the length of its backbone is b cm, then itsheight can be modelled by the function
h(b) = 0.721b^0.633.
(Note: These type of relationships are studied in the subject of allometry. This equation is referred to as an allometric equation.

(a) Find the derivative of h with respect to b.

(b) Suppose that the length of the backbone is related to the age x of the dinosaur. Thus, we will consider the length of the backbone b(x) as a differentiable function in x. This, in turn, implies that we may consider the dinosaur’s height h(x) as a function of x. Determine dh/dx

(c) We now wish to consider the relative growth rates of the height and backbone. Find a positive constant k such that

[1/h(x) * (dh/dx)] = [K*(1/b(x)) * (db/dx)]

(d) Based on the value k found in part c), which grows faster, the dinosaur’s height or backbone? Explain.

Attempt at a solution:

for part a) h'(b) = 0.456b^(-0.367)

b) h'(x) = 0.456(b(x))^ (-0.367) b'(x)

c) I got K= 0.633, is that right?

 

[mmm4444bot]

2. A dinosaur’s height is related to the length of its backbone.In particular, if the length of its backbone is b cm, then itsheight can be modelled by the function

h(b) = 0.721b^0.633.

(b) Suppose that the length of the backbone is related to the age x of the dinosaur. Thus, we will consider the length of the backbone b(x) as a differentiable function in x. This, in turn, implies that we may consider the dinosaur’s height h(x) as a function of x. Determine dh/dx

Attempt at a solution:

for part a) h'(b) = 0.456b^(-0.367)

I don't know what part b is asking for, can someone clarify and point me in the right direction?

Your derivative in part (a) is correct.

Part (b) makes a change in the definition of function h.

The base b is now itself a function. They do not give you any definition for this function b(x); you will need to express its derivative simply as b'(x) or db/dx.

So, the new function h in part (b) is:

h(x) = 0.721*b(x)^0.633

Use the Chain Rule, when determining h'(x)

 

[as_xoxo]

Your derivative in part (a) is correct.

Part (b) makes a change in the definition of function h.

The base b is now itself a function. They do not give you any definition for this function b(x); you will need to express its derivative simply as b'(x) or db/dx.

So, the new function h in part (b) is:

h(x) = 0.721*b(x)^0.633

Use the Chain Rule, when determining h'(x)

Thank you!

I did that and obtained h'(x) = 0.456(b(x)^-0.367 b'(x)

and when solving for part c, K ends up to be 0.633, does that make sense?