💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Get the answer to your homework problem.

Try Numerade Free for 7 Days

Like

Report

Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function $ f(r) = 2 \sqrt {Dr}, $ where $ r $ is the reproductive rate of individuals and $ D $ is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate $ r $ and explain its meaning.

$f^{\prime}(r)=\sqrt{\frac{D}{r}}$

00:28

Amrita B.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Derivatives

Differentiation

Campbell University

Oregon State University

University of Nottingham

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:11

Invasive species often dis…

12:05

Waves of Arbitrary Shape. …

02:02

Wave equation Traveling wa…

02:40

Bacteria population geneti…

01:44

Population Growth When a h…

01:41

Traveling waves (for examp…

02:33

42. Predator-prey dynamics…

01:49

03:14

The velocity of a surface …

09:46

The $ \textit{diffusion eq…

all right, so this function represents the speed of the wave of the invasive species. Colonising as a function of our the reproductive rate and D is just some constant, and we want to find the derivative. The derivative is going to represent the acceleration of the wave as a function of the reproductive weight rate, because the derivative of speed would be acceleration. So before we find the derivative, let's change the equation to to Times Square root D That should be a two to times square root d. That's the constant part of it. Times are to the 1/2 that's the variable part of it. Now let's find the derivative. So F crime of our is the constant to square root D multiplied by the derivative of art. Of the 1/2 which would be 1/2 are to the negative 1/2 so we can simplify that by multiplying the two in the 1/2. And then we can think of are to the negative 1/2 power as one over square root are and then we might as well combine those into the same square roots. So we have the rate of change of the speed is the square root of D over our

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Invasive species often display a wave of advance as they colonize new areas.…

Waves of Arbitrary Shape. (a) Explain why any wave described by a function o…

Wave equation Traveling waves (for example, water waves or electromagnetic w…

Bacteria population genetics Suppose there are two bacterial strains 1 and $…

Population Growth When a hardy new species is introduced into an area, the p…

Traveling waves (for example, water waves or electromagnetic waves ) exhibit…

42. Predator-prey dynamics In Chapter 7 we study a modelfor the populati…

The velocity of a surface wave on the ocean is given by $v=\sqrt{\frac{g \la…

The $ \textit{diffusion equation} $ $$ \dfrac{\partial c}{\partial t} = D \d…