Problem 1 - Active Calculus Multivariable

Question

Evaluate a function. Evaluate the function at the specified points. $$ f(x, y)=x+y x^{4},(-1,3),(-4,2),(-1,-3) $$ At (-1,3):_______ At (-4,2):_______ At (-1,-3):_______

Step-by-Step Solution

Answer

At (-1,3): 2 At (-4,2): 508 At (-1,-3): -4

Step 1: Identify the function

The function given is \(f(x, y) = x + yx^4\), where x and y are our input variables.

Step 2: Evaluate the function at point (-1,3)

We are given the point \((-1, 3)\). Hence, \(x=-1\) and \(y=3\). Plug in the values into the function: \[f(-1, 3) =-1 +3(-1)^4\] \[f(-1, 3) = -1+3\] \[f(-1, 3) = 2\] So, at the point \((-1, 3)\), the function has a value of 2.

Step 3: Evaluate the function at point (-4,2)

We are given the point \((-4, 2)\). Hence, \(x=-4\) and \(y=2\). Plug in the values into the function: \[f(-4, 2) =-4 +2(-4)^4\] \[f(-4, 2) = -4+2(256)\] \[f(-4, 2) = -4+512\] \[f(-4, 2) = 508\] So, at the point \((-4, 2)\), the function has a value of 508.

Step 4: Evaluate the function at point (-1,-3)

We are given the point \((-1, -3)\). Hence, \(x=-1\) and \(y=-3\). Plug in the values into the function: \[f(-1, -3) =-1 -3(-1)^4\] \[f(-1, -3) = -1-3\] \[f(-1, -3) = -4\] So, at the point \((-1, -3)\), the function has a value of -4.

Step 5: Write the results

We have found the function values at the specified points: At (-1,3): 2 At (-4,2): 508 At (-1,-3): -4

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for specific inputs. It means substituting the given coordinate values (in this case, points (-1, 3), (-4, 2), and (-1, -3)) into the function provided, which is \(f(x, y) = x + yx^4\). Evaluating a function helps in understanding how the function behaves at different input values and can be particularly important when analyzing real-world problems.

• Start by identifying the function.
• Replace the variables \(x\) and \(y\) with the respective values from the coordinate points.
• Simplify the expression to get the function's output.

Through these evaluations, you gain insight into the relationship between input variables and the function's output. This is a fundamental concept in calculus and algebra.

Coordinate Points

Coordinate points are pairs of numbers that define a specific location on a plane. In the context of a function with two variables \((x, y)\), each coordinate point gives us specific values for these variables. For example, the point (-1, 3) means \(x = -1\) and \(y = 3\).

• Coordinate points are written in the format \((x, y)\).
• The first number in the pair corresponds to the \(x\)-value.
• The second number in the pair corresponds to the \(y\)-value.

Using coordinate points, we can plug these values into a multivariable function to evaluate it at that particular point. This process helps us determine how the output changes with different inputs.

Polynomial Functions

Polynomial functions are algebraic expressions involving variables raised to whole number powers and coefficients. The function given in the exercise, \(f(x, y) = x + yx^4\), is a polynomial function.

• A term like \(yx^4\) includes a variable \(x\) raised to the 4th power and is multiplied by another variable \(y\).
• Polynomials can have one or multiple terms. Each term involves multiplying constant coefficients and variables raised to powers.
• They can represent simple lines, complex curves, or connections across multiple variables.

Polynomial functions are widely used due to their simple structure and ability to model various phenomena. Here, substitution of particular coordinate point values shows the impact of the function across different inputs.

Plug and Chug Method

The 'Plug and Chug' method is an informal but widely used approach in math for evaluating expressions. It involves simply substituting values into equations and calculating the result step-by-step.

• Identify what the equation requires for its variables.
• Substitute given value(s) in place of the variables from the coordinate points.
• Systematically simplify and compute the final result.

With the function \(f(x, y) = x + yx^4\), for example, you systematically replace \(x\) and \(y\) with values from each given point, such as \((-1, 3)\), then perform the mathematics to reach the outcome.Practicing this approach can help build confidence and understanding when dealing with functions, enhancing your problem-solving skills.