Problem 18 - The Calculus with Analytic Geometry

Question

\(y=\frac{5 x-2}{x^{2}+1} ; x=(2 t-1)^{2}\)

Step-by-Step Solution

Answer

The substitution yields: \[ y = \frac{20t^2 - 20t + 3}{(2t-1)^4+1} \]

Step 1 - Identify the Given Equations

We have two equations: 1) The function is given as: \[ y = \frac{5x - 2}{x^2 + 1} \] 2) The variable substitution for x is given as: \[ x = (2t - 1)^2 \]

Step 2 - Substitute x in the Function

Substitute the expression for x into the function for y. Since \( x = (2t - 1)^2 \), we get: \[ y = \frac{5( (2t - 1)^2 ) - 2}{( (2t - 1)^2 )^2 + 1} \]

Step 3 - Simplify the Denominator

Simplify the denominator: \[ ( (2t - 1)^2 )^2 = (2t - 1)^4 \] So the denominator becomes: \[ (2t - 1)^4 + 1 \]

Step 4 - Simplify the Numerator

Simplify the numerator: \[ 5 ( (2t - 1)^2 ) - 2 = 5 (4t^2 - 4t + 1) - 2 \] which simplifies to \[ 20t^2 - 20t + 5 - 2 = 20t^2 - 20t + 3 \]

Step 5 - Combine the Results

Now, combine the simplified numerator and denominator into the function for y: \[ y = \frac{20t^2 - 20t + 3}{(2t-1)^4+1} \]

Key Concepts

Function substitution

In calculus, function substitution is a technique where one function is substituted into another. This is very useful when dealing with composite functions or complex expressions.

Consider the given exercise where we substitute a function for the variable \( x \).
Here, the function for \( y \) is given by:
\ \ \, \ \
\ \ y = \ \frac{5x \ - \ 2}{x^2 \ + \ 1} \ (1) \ \
And the substitution for \( x \) is provided as:
\ \
x \= (2t \ - \ 1)^2 \ (2) y \
The goal is to replace \( x \) in equation (1) with the expression given in equation (2). This process is systematic and involves substituting the given expression step by step.

This method can significantly simplify complex calculus problems.

Simplification

Simplification involves reducing expressions to their simplest form. This can often include combining like terms or reducing fractions. In our exercise, after substituting for \( x \), the expression for \( y \) becomes more complicated.

To simplify this expression, focus on the numerator and denominator separately. For the numerator:
\ \ y = \ \frac {5( (2t \ - \ 1)^2 \ ) - \ 2}{( (2t \ - \ 1)^2 \ )^2 \ + \ 1} (2)}

This step is performed by squaring \( 2t - 1 \) and multiplying by 5, then simplifying further.

Algebraic manipulation

Algebraic manipulation is essential for solving calculus problems. It includes expanding expressions, factoring, combining like terms, and reducing complex fractions.

In our problem, expanding \( (2t - 1)^2 \) and then performing arithmetic operations correctly is crucial.

For the numerator: \ \ [[ 5( (2t \ - \ 1)^2 ) \ - \ 2 starts with: \ \
y = \5 (4t^2 \ - \ 4t \ + \ 1) \ - \ 2]}= [[\20t + \ -20 \ + \ \}}
\ 3]]

\ (The above
term expression for exponential can already

Rational functions

A rational function is a function that is the ratio of two polynomials. In our exercise, \( y \) is defined as such a function. Rational functions can have intricate behavior due to their polynomial nature in the numerator and denominator.

The given function for \( y \) is:

\ y \ = \ \frac{5x \ - \ 2}{x^2 \ + \ 1}

After substitution and simplification, the rational function in terms of \( t \ ) is: \ \ text {respectively\20(2t^2^{-20\,t + \}} (2t\) completes
\ \ (substitution more manageable allows representation efficient Understanding substitution to exercises calculus complex