91Ó°ÊÓ

Calculus is a branch of mathematics that focuses on change and motion. It primarily deals with two fundamental concepts: differentiation and integration. In this context, we are interested in differentiation, one of the core building blocks of calculus.

By studying calculus, we can understand how functions behave at any given point. It allows us to calculate rates of change and understand the curvature of functions. In our exercise, we use calculus to explore the behavior of the function \( f(t) = t^4 - 3t^2 + 5t \).

Studying calculus provides powerful tools to solve real-world problems. For instance, it's used in physics to model motion and in economics to find optimum solutions.

Differentiation is the process of finding the derivative of a function. A derivative represents how a function changes as its input changes. In simple terms, the derivative gives us the slope of the tangent line to the function at any point.

In our exercise, we find the first and second derivatives of the function \( f(t) = t^4 - 3t^2 + 5t \).

- The first derivative, \( f'(t) \), is calculated by taking the derivative of each term of the polynomial individually:

• The derivative of \( t^4 \) is \( 4t^3 \).
• The derivative of \( -3t^2 \) is \( -6t \).
• The derivative of \( 5t \) is \( 5 \).

This gives us \( f'(t) = 4t^3 - 6t + 5 \).

- To find the second derivative, \( f''(t) \), we differentiate \( f'(t) \) again:

• The derivative of \( 4t^3 \) is \( 12t^2 \).
• The derivative of \( -6t \) is \( -6 \).
• The derivative of the constant \( 5 \) is \( 0 \).

Thus, we obtain \( f''(t) = 12t^2 - 6 \).

Polynomials are mathematical expressions consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. They are one of the simplest types of functions and are easy to work with when it comes to differentiation.

In our exercise, we work with the polynomial \( f(t) = t^4 - 3t^2 + 5t \). Each term in a polynomial can be differentiated easily because of the simple power rule for differentiation. The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \). This makes it straightforward to find the derivatives of polynomials.

Polynomials form the basis of more complex functions, and understanding how to differentiate them is crucial in calculus. They allow us to model and solve a wide range of practical problems in science and engineering.