The velocity (in centimeters per second) of blood r cm from the central axis of an artery is giv

step 1 According to the question, the velocity of blood R -centimeter from the central axis of an arter

The velocity (in centimeters per second) of blood r cm from...

The velocity (in centimeters per second) of blood $r \mathrm{cm}$ from the central axis of an artery is given by $$v(r)=k\left(R^{2}-r^{2}\right)$$ where $k$ is a constant and $R$ is the radius of the artery (see the accompanying figure). Suppose $k=1000$ and $R=0.2 \mathrm{cm} .$ Find $v(0.1)$ and $v^{\prime}(0.1),$ and interpret your results.

The velocity (in centimeters per second) of blood $r \mathrm{cm}$ from the central axis of an artery is given by
$$v(r)=k\left(R^{2}-r^{2}\right)$$
where $k$ is a constant and $R$ is the radius of the artery (see the accompanying figure). Suppose $k=1000$ and $R=0.2 \mathrm{cm} .$ Find $v(0.1)$ and $v^{\prime}(0.1),$ and interpret your results.

Key Concepts

Function Evaluation

This key concept involves substituting a specific value into a function to determine the output at that point. In mathematical modeling, evaluating a function at a given input allows us to understand the behavior or state of a system at a particular condition, making it essential for analyzing physical systems like velocity distributions.

Differentiation

Differentiation is the process of computing the derivative of a function, which quantifies how the function's output changes with respect to changes in the input. This is a foundational concept in calculus that helps in determining rates of change and slopes, and is crucial for understanding dynamic systems.

Interpretation of the Derivative

The derivative, obtained through differentiation, represents the instantaneous rate of change of the function with respect to its variable. In a physical context, such as fluid flow, it provides insight into how rapidly the velocity changes at a specific point, which can be critical for understanding forces and shearing effects in the medium.

Mathematical Modeling in Fluid Dynamics

In fluid dynamics, mathematical modeling involves using functions to represent physical quantities like velocity, pressure, and flow profiles. Parabolic velocity profiles, for example, are typical in laminar flow conditions within pipes or arteries. This concept underscores the role of applied mathematics in predicting and interpreting the behavior of physical systems.

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