Calculator Techniques FOR BOARD EXAM - Evaluating Functions and Simplifying Expressions 10 EXAMPLES

This paragraph introduces the video's focus on calculator techniques essential for engineering board exam preparation. The speaker emphasizes the importance of these techniques for passing the math section of the exam, particularly for mechanical, electrical, and electronics engineering students. The tutorial is designed to cover various mathematical topics applicable to calculator use, starting with algebra and progressing to more complex subjects like derivatives, integrals, differential equations, and analytic geometry. The speaker clarifies that calculator techniques are only applicable to multiple-choice questions.

The speaker provides a step-by-step guide on how to use a calculator to evaluate functions, using the function f(x) = 2x^2 - 5x + 3 as an example. They explain the process of turning on the calculator, inputting the equation, and evaluating it for specific values of x (in this case, x = 2 and x = 4). The explanation includes details about using the 'alpha' function to denote outputs and the importance of using a calculator that is allowed in the exam setting, such as the fx-570ES PLUS mentioned.

The paragraph explains how to use the Remainder Theorem to find the remainder of a polynomial function when divided by a linear factor. The speaker illustrates this with the function 2x^3 + 3x^2 - 2x + 25, showing how to set up the calculator to find the remainder when the function is divided by x + 3. The process involves equating x + 3 to zero to find the value of x for which f(x) will be the remainder, which in this case is x = -3, leading to a remainder of 4.

The speaker discusses using calculator techniques to identify factors of the polynomial 3x^3 + 2x^2 - 32, given multiple-choice options. They explain the Factor Theorem, which states that if x - k is a factor of a polynomial, then f(k) should equal zero. The process involves inputting the polynomial into the calculator and evaluating it for each potential factor's value to determine if the remainder is zero, thus confirming it as a factor.

This section describes how to calculate the difference between f(x+1) and f(x) for the function f(x) = 10^(x+1). The speaker suggests using a calculator to compute the numerical values of f(3) and f(2) and then subtracting the latter from the former to find the difference. The correct answer is identified through the process of elimination using multiple-choice options, with the correct answer being 900, corresponding to option D.

The paragraph explains the process of evaluating a function with two variables, f(x, y) = 4x^3 + 3x^2y - 5xy^2 + y^3, using a calculator. The speaker demonstrates how to input the function into the calculator, assign values to x and y, and then evaluate the function for those values. The example given uses x = 1 and y = 2, resulting in an answer of -2, which corresponds to option C.

The speaker presents a method for simplifying the expression (x^4 - y^4) / (x^4 - 2x^2y^2 + y^4) using a calculator. They suggest assigning values to x and y, inputting the expression into the calculator, and then evaluating it to find the simplified result. The example uses x = 2 and y = 3, leading to a simplified result that matches one of the multiple-choice options, identified as option B.

This section covers the simplification of an expression involving exponential terms with variables x, y, and z. The speaker advises assigning values to these variables and inputting the expression into the calculator to find the simplified result. The example uses x = a, y = b, z = c with values a = 2, b = 3, and c = 4, resulting in a simplified expression that matches one of the multiple-choice options, identified as option A.

The paragraph explains how to solve for x in an equation involving exponential terms with variable n. The speaker demonstrates the process of simplifying the equation by replacing 4^n with x and then solving for x using different values of n. The example shows that regardless of the value used for n, the result for x remains the same, indicating the solution for x in terms of n.

The speaker simplifies the expression a^(-2) * b^6 / (a^2 * b^(-1)) using a calculator with assigned values for a and b. The example uses a = 2 and b = 3, resulting in a simplified result of 2187/64. The speaker then compares this result with multiple-choice options to identify the correct answer, which is option C.

The final paragraph demonstrates solving for the variable a in the equation a = 64^x * 4^y using a calculator. The speaker shows how to input the equation into the calculator, assign values to x and y, and compute the numerical value of a. The example uses x = 2 and y = 3, leading to a value for a that matches one of the multiple-choice options, identified as option D.