Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{t \rightarrow 3}\left(4 t^{2}-2 t+1\right)\)

Step-by-Step Solution

Answer

The limit is 31.

Step 1: Evaluate the function at t=3

Begin by substituting the value \(t = 3\) into the function: \(4(3)^2 - 2(3) + 1\) .

Step 2: Calculate the function's value

Now, perform the mathematical operations in this expression: \[4(3)^2 - 2(3) + 1 = 4(9) - 6 + 1 = 36 - 6 + 1\]

Step 3: Simplify the expression

Simplify the expression to calculate the limit: \[36 - 6 + 1 = 30 + 1 = 31\] So, \(\lim _{t \rightarrow 3}\left(4 t^{2}-2 t+1\right) = 31\).

Key Concepts

Limits

Limits play a crucial role in calculus, especially when dealing with functions that seem complicated or undefined at certain points. A limit essentially helps us understand what a function approaches as the input gets closer to some value. In the exercise given, we are asked to find \( \lim_{t \to 3} (4t^2 - 2t + 1) \).This means we are interested in what the function \(4t^2 - 2t + 1\) is approaching as \(t\) gets very close to 3.
When we say "approaching," it is important to note that the variable doesn't necessarily reach this value. Instead, we look at values increasingly closer to the point of interest. Limits help provide a bridge to find function values that aren't immediately obvious or when the function exhibits undefined behavior at a point. Understanding limits is fundamental to grasping more advanced concepts like derivatives and integrals in calculus.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers, often combined with coefficients. The function in our exercise,\(4t^2 - 2t + 1\), is a quadratic polynomial. This means it has a variable raised to the second power (the highest degree is 2), and its graph forms a parabola.
Some features of polynomial functions include:

• They are smooth and continuous for all real numbers.
• You can differentiate and integrate polynomials easily.
• As the degree increases, polynomials can start having more turning points and intersections with the x-axis.
• They do not have breaks, jumps, or asymptotes like some other function types.

Polynomials are integral to many areas of mathematics and are used to model a variety of physical systems, so understanding their properties is key to solving complex calculus problems.

Substitution Method

The substitution method is a straightforward technique used to find limits. It involves replacing the variable in the function with the value it approaches. This method is possible when the function is continuous at the point of interest. In our example, we replaced \(t\) with 3 in the polynomial. This calculated the value directly:1. Substitute \(t = 3\) into \(4t^2 - 2t + 1\).2. Simplify: \(4(3)^2 - 2(3) + 1 = 36 - 6 + 1 = 31\).
However, this method only works seamlessly when the polynomial doesn't result in undefined forms like division by zero. It is a simple strategy but requires caution. If the substitution leads to complications, other techniques like factoring or L'Hôpital's Rule might be necessary. In calculus, mastering substitution can be incredibly helpful for solving a variety of functions and simplifies the computation of limits.