Problem 4 - Calculus for Biology and Medicine

Question

Recall that the acceleration \(a(t)\) of a particle moving along a straight line is the instantaneous rate of change of the velocity \(v(t) ;\) that is, $$ a(t)=\frac{d}{d t} v(t) $$ Assume that \(a(t)=32 \mathrm{ft} / \mathrm{s}^{2}\). Express the cumulative change in velocity during the interval \([0, t]\) as a definite integral, and compute the integral.

Step-by-Step Solution

Answer

The cumulative change in velocity over the interval \\[0, t\\] is \\32t \, \text{ft/s}\\.

Step 1: Express the Cumulative Change as an Integral

To find the cumulative change in velocity over a time interval \[0, t\], integrate the acceleration function \a(t)\ with respect to time. The definite integral is expressed as: \[\int_{0}^{t} a(t) \, dt\] where \a(t) = 32 \, ext{ft/s}^2\. Therefore, the integral becomes: \[\int_{0}^{t} 32 \, dt\].

Step 2: Compute the Definite Integral

To compute the integral \[\int_{0}^{t} 32 \, dt\], apply the power rule of integration. The integral of a constant is the constant times the variable of integration. Thus, \[\int_{0}^{t} 32 \, dt = \left[32t\right]_{0}^{t}\].

Step 3: Apply the Limits of Integration

Substitute the upper and lower limits of integration into the result from Step 2. Evaluate the expression: \[32t\right]_{0}^{t} = 32t - 32(0) \]. Simplifying gives \32t\, which is the cumulative change in velocity over the interval \[0,t\].

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus, representing the accumulation of quantities, such as areas under curves or total quantities over an interval. Think of it like adding up tiny pieces to find a whole. When dealing with definite integrals, we focus on a specific range or interval.

• The limits of integration define the start and end of the interval.
• The process of integration accumulates values from the lower limit to the upper limit.

Integrals are often used to compute totals that cannot be easily measured directly. For example, when integrating acceleration over time, you're essentially collecting all the small changes in velocity. The key to solving definite integrals often lies in understanding and applying basic integration rules. These rules allow us to transform integral expressions into solvable equations. Once you have the integral, evaluating it is just about plugging in the limits.

Rate of Change

The rate of change is a crucial concept in calculus, specifically referring to how a quantity changes over time. In the context of motion, it helps us understand how fast something is changing, like speed or speed-up (acceleration).

• Velocity is the rate of change of position – it tells us how fast an object is moving and in which direction.
• Acceleration is the rate of change of velocity – it indicates how fast the velocity itself is changing.

When expressed mathematically, rates of change provide a snapshot of what's happening at any instant. This is why we use derivatives to find these rates. For instance, the derivative of velocity with respect to time gives us the acceleration. This is useful not only for understanding movement but also for many real-world applications such as predicting trends and modeling physical systems.

Velocity and Acceleration

The relationship between velocity and acceleration is central in physics and calculus, as it describes an object's motion through space over time.
Velocity is a vector quantity that indicates speed and direction of motion, while acceleration is the change in velocity.

• Acceleration occurs if there's a change in the speed or direction of the velocity.
• Even if the speed is constant, changing direction means there's acceleration.

To calculate velocity using acceleration, we take the integral of the acceleration function over a given interval. This accumulates all the little increases and decreases in speed to find the total change:\[\int_0^t a(t) \; dt\]Through integration, we transition from acceleration to finding velocity. This process highlights calculus's power: using derivatives and integrals to understand changing quantities. Calculus allows us to predict how objects move, making it a highly useful tool in both science and engineering.