How Can Knowledge Of Calculus Optimise Your Workout Routine

Physical fitness is an important aspect of a healthy lifestyle. It is essential to have a regular workout routine to keep your body healthy and fit. However, sometimes it can be challenging to determine the optimal workout routine that is right for you. That's where calculus comes handy. Calculus can help you optimise your workout routine by providing a mathematical understanding of how your body works during exercise. So let’s find out how calculus can be used to optimise your physical workout routine.

First, let's start with the basics. Calculus is a branch of mathematics that deals with the study of rates of change and how things change over time. When it comes to fitness, calculus can help us understand the changes that occur in our bodies during exercise. Specifically, calculus can help us optimise our workouts by understanding the relationship between variables such as heart rate, oxygen consumption, and calorie burn.

Fitness Parameters

Now, consider these fitness parameters that can be used to assess our fitness level..

Basal Metabolic Rate (BMR)

BMR, Basal Metabolic Rate is the amount of energy that the body needs to function at rest, meaning the energy required to maintain basic body functions such as breathing, circulation, and digestion. BMR can vary from person to person based on factors such as age, sex, weight, height, and body composition. Knowing your BMR can help you determine the number of calories your body requires to maintain its basic functions.

BMR can be calculated using the Harris-Benedict equation, which takes into account an individual's age, sex, weight, and height. Here is the formula for calculating BMR:

For men: BMR = 88.362 + (13.397 x weight in kg) + (4.799 x height in cm) - (5.677 x age in years)

For women: BMR = 447.593 + (9.247 x weight in kg) + (3.098 x height in cm) - (4.330 x age in years)

For example, if a 30-year-old woman weighs 65 kg and is 165 cm tall, her BMR would be calculated as follows:

BMR = 447.593 + (9.247 x 65) + (3.098 x 165) - (4.330 x 30)

BMR = 1,394.765 calories per day

This means that her body requires approximately 1,394.765 calories per day to maintain basic functions at rest. However, it is important to note that this is only an estimate, and other factors such as physical activity level and body composition can also affect an individual's overall energy needs.

Heart Rate

Resting Heart Rate (RHR) refers to the heart rate when an individual is at rest, usually measured upon waking up in the morning. To measure RHR, an individual can use their index and middle finger to place on either their radial artery on the wrist or carotid artery in the neck and count the number of beats in 10 seconds, then multiply that number by 6.

On the other hand, Maximum Heart Rate (HRmax) refers to the highest heart rate an individual can achieve through exercise stress. This is often used as a guideline for determining exercise intensity and creating workout programs

Maximum Heart Rate (HRmax) = 220 - Individual's age.

Heart Rate Reserve (HRR) = Maximum Heart Rate (HRmax) - Resting Heart Rate (RHR)

Target Heart Rate (THR) at a specific exercise intensity level.

Target Heart Rate = (HRR / desired exercise intensity percentage) + RHR

BMI (Body Mass Index)

Body Mass Index (BMI) is a measurement of a person's weight in relation to their height. It is a widely used method to assess whether a person has a healthy body weight, or is overweight or underweight. BMI is calculated by dividing a person's weight (in kilograms) by the square of their height (in metres).

The formula for calculating BMI is

BMI = weight (kg) / (height (m))²

For example, if a person weighs 70 kilograms and their height is 1.75 metres, their BMI would be calculated as follows:

BMI = 70 / (1.75)²

BMI = 22.9

Fitness categories according to BMI range

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Optimise Physical Workout

By using calculus, we can gain a better understanding of how our bodies respond to exercise and make informed decisions about how to adjust our workout routine to meet our fitness goals. There are multiple concepts such as derivatives and integrals in calculus. Let's apply these concepts to optimise our fitness routine.

Derivatives can be used to optimise a workout routine by analysing the rate of change of different fitness parameters, such as heart rate, and adjusting the intensity of the workout accordingly. Here are a few examples of how derivatives can be applied in real-life scenarios:

Adjusting The Intensity Of A Cardio Workout

1. Suppose you're performing a cardio workout, and you want to optimise your heart rate to achieve your fitness goals. By calculating the derivative of your heart rate with respect to time, you can determine how quickly your heart rate is changing. If it's not increasing fast enough to meet your goals, you can adjust the intensity of your workout by increasing the resistance or incline on your machine or running faster. Mathematically, this can be expressed as:

dHR/dt = (HR2 - HR1)/(t2 - t1)

Where HR1 and HR2 are your heart rates at two different times t1 and t2, respectively. By calculating the derivative of this equation, we can determine the rate of change of heart rate over time.

Optimising Weight Lifting Repetitions

2. Suppose you're lifting weights, and you want to optimise the number of repetitions you perform to build muscle mass. By calculating the derivative of the force you're exerting with respect to time, you can determine the rate at which you're fatiguing. If you're fatiguing too quickly and not reaching your desired number of repetitions, you can adjust the weight or rest time between sets to optimise your workout. Mathematically, this can be expressed as:

dF/dt = (F2 - F1)/(t2 - t1)

Where F1 and F2 are the forces you're exerting at two different times t1 and t2, respectively.

These are just a few examples of how derivatives can be used to optimise a workout routine. Now lets us understand another key concept of calculus i.e integral.

Integral calculus can be used to optimise a workout by analysing the total work done during a workout session. The concept of work in physics is defined as the product of force and distance. In the case of a workout, this can be interpreted as the product of the force exerted by the body and the distance covered during the workout.

To illustrate this concept, let's consider the example of running on a treadmill. Suppose a person runs at a constant speed of 10 km/hour for 30 minutes on a treadmill that has an incline of 5 degrees. The total distance covered by the person during this workout can be calculated using the formula:

Distance = Speed x Time

Distance = 10 km/hour x 0.5 hour (30 minutes converted to hours)

Distance = 5 kilometres

Next, we can calculate the total work done by the person during this workout session using the formula:

Work = Force x Distance

Work = (mass x gravity x incline angle) x Distance

Assuming the person has a mass of 70 kg and the acceleration due to gravity is 9.81 m/s^2, the force exerted by the person can be calculated as

Force = mass x gravity

Force = 70 kg x 9.81 m/s^2

Force = 686.7 N

Substituting this value into the work equation and converting the distance to metres, we get:

Work = 686.7 N x 5,000 metres x sin(5 degrees)

Work = 28,087 Joules

This value represents the total exertion taken by the person during their workout. By tracking this value over time, we can optimise the workout by adjusting the speed, incline, and duration of the workout to achieve our fitness goals. For example, increasing the incline or duration of the workout will increase the total work done, leading to greater improvements in fitness.

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