Calculus, often described as the language of change, is a branch of mathematics that deals with the study of rates of change, areas, volumes, and other concepts related to varying quantities. One of the most intriguing aspects of calculus is the ability to solve complex problems through the use of derivatives and integrals. In this article, we will delve into a calculus question that challenges the understanding of limits and their application in solving real-world problems.>
Consider the following calculus question: “A particle moves along a straight line such that its position, s(t), is given by the function s(t) = 3t^2 – 4t + 5, where t is measured in seconds and s(t) is measured in meters. Find the velocity of the particle at time t = 2 seconds.” This question requires the application of calculus concepts to determine the rate of change of the particle’s position with respect to time.>
Before we proceed to solve the calculus question, let’s first understand the concept of velocity in calculus. Velocity is defined as the derivative of the position function with respect to time, which represents the rate at which the position of an object changes over time. To find the velocity of the particle at t = 2 seconds, we need to calculate the derivative of the position function, s(t), with respect to time, t.>
Let’s start by finding the derivative of the position function, s(t) = 3t^2 – 4t + 5. The derivative of a function represents the slope of the tangent line to the curve at a given point. To find the derivative of s(t), we will apply the power rule, which states that the derivative of t^n is nt^(n-1).>
Using the power rule, we can find the derivative of each term in the position function:
ds/dt = d(3t^2)/dt – d(4t)/dt + d(5)/dt
ds/dt = 23t^(2-1) – 14t^(1-1) + 0
ds/dt = 6t – 4
Now that we have the derivative of the position function, we can find the velocity of the particle at t = 2 seconds by substituting t = 2 into the derivative:
v(2) = 62 – 4
v(2) = 12 – 4
v(2) = 8
Thus, the velocity of the particle at t = 2 seconds is 8 meters per second. This solution demonstrates the power of calculus in solving real-world problems, as it allows us to determine the rate of change of a quantity in various scenarios.>
