Problem 26 - Calculus

Question

Involve numbers \(\int_{0}, f_{1}, f_{2}, \ldots\) and their differences \(v_{j}=f_{j}-f_{j-1}\), They give practice with subscripts \(0, \ldots, j\). If \(f(t)=t^{2}+t,\) compute \(f(99)\) and \(f(101)\). Between those times, what is the increase in \(f\) divided by the increase in \(t\) ?

Step-by-Step Solution

Answer

The increase in \( f \) divided by the increase in \( t \) is 201.

Step 1: Evaluate f(99)

Substitute \( t = 99 \) into the function \( f(t) = t^2 + t \).\[f(99) = 99^2 + 99 = 9801 + 99 = 9900\]

Step 2: Evaluate f(101)

Substitute \( t = 101 \) into the function \( f(t) = t^2 + t \).\[f(101) = 101^2 + 101 = 10201 + 101 = 10302\]

Step 3: Calculate the Difference in f

Find the difference between \( f(101) \) and \( f(99) \).\[f(101) - f(99) = 10302 - 9900 = 402\]

Step 4: Calculate the Difference in t

Find the difference between \( t = 101 \) and \( t = 99 \).\[101 - 99 = 2\]

Step 5: Compute the Increase in f Divided by the Increase in t

Divide the difference in \( f \) by the difference in \( t \) to find the average rate of change.\[\frac{f(101) - f(99)}{101 - 99}= \frac{402}{2} = 201\]

Key Concepts

Understanding Polynomial Functions

Polynomial functions are algebraic expressions that involve terms made up of variables raised to whole number powers, often combined with constants. They resemble this general form: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Here, each \(a_n\) is a coefficient, and \(x\) is the variable. A simple example of a polynomial function is \(f(t) = t^2 + t\), which belongs to up to degree 2 since the highest power of \(t\) is 2. Polynomial functions can be classified by degree:

• Constant polynomials, where degree = 0 (e.g., \(f(x) = 5\))
• Linear polynomials, where degree = 1 (e.g., \(f(x) = 2x+1\))
• Quadratic polynomials, where degree = 2 (e.g., \(f(x) = x^2 + 3\))

Understanding the degree and structure of polynomial functions helps in analyzing how they behave for different values of \(x\).

Decoding Subscripts

Subscripts are small numbers or characters written at the lower right of a variable or letter. They serve as identifiers to distinguish between similar items or represent sequences. In the context of functions and sequences, subscripts can be used to denote:

• Different function outputs like \(f_0, f_1, f_2\).
• Elements in a sequence or different versions of the same variable.

For example, in the exercise, different function evaluations like \(f_{99}\) represent function values at different input points. They provide a convenient way to track changes or differences between successive function values or other elements.

Explaining the Difference Quotient

The difference quotient is a central concept in calculus that represents the average rate of change of a function over a certain interval. It is expressed as:\[\frac{f(b) - f(a)}{b - a}\]This formula considers how much the function output changes (∆f) over a change in input (∆x). It's essential for finding slope or rate of change in line segments of polynomial functions.In the exercise case, the difference quotient tells how fast the function changes between \(t = 99\) and \(t = 101\). Calculating this quotient helps us understand the behavior or trend of the polynomial function over that interval. It is a foundational step towards understanding derivatives, which gauge change in functions over infinitesimally small intervals.

Function Evaluation Made Simple

Function evaluation involves substituting a specific value into a function and calculating the outcome. For a polynomial like \(f(t) = t^2 + t\), evaluating means plugging in a number for \(t\), then performing arithmetic operations to find the result.Consider evaluating \(f(99)\):

• Replace \(t\) with 99, following the function definition: \(f(99) = 99^2 + 99\)
• Calculate each term: \(99^2 = 9801\) and add 99 to get \(9900\).

This process resembles solving a puzzle, where you systematically replace variables with numbers, simplify step by step, and arrive at a precise answer. It's critical for understanding how functions behave and change with different inputs.