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

This paragraph introduces the video's focus on calculator techniques for engineering board exams, emphasizing their importance for students across various engineering disciplines. The speaker shares personal experience, highlighting how these techniques were crucial for passing the math section of the board exam. The video aims to cover all applicable math topics for calculator use, starting with algebra, evaluating functions, and simplifying expressions, and will continue to derivatives, integrals, differential equations, and analytic geometry. The speaker clarifies that these techniques are only applicable to multiple-choice questions.

The speaker provides a step-by-step tutorial on how to use a calculator to evaluate functions, using a specific function f(x) = 2x^2 - 5x + 3 as an example. They explain the process of turning on the calculator, inputting the equation correctly by using the alpha function for the variable x, and then evaluating the function for specific values of x (in this case, x=2 and x=4). The explanation includes navigating the calculator's interface, such as using alpha for function notation and shift for additional operations, and concludes with solving the problem using the calculator to find the correct multiple-choice answer.

This section discusses the application of the Remainder Theorem to find the remainder of a polynomial function when divided by a linear factor. The speaker illustrates how to use the calculator to find the remainder of the function 2x^3 + 3x^2 - 2x + 25 when divided by x + 3. They explain the process of equating the divisor to zero to find the value of x (in this case, x = -3) and then inputting this value into the function to calculate the remainder, which turns out to be 4, corresponding to a multiple-choice answer.

The speaker explains how to use calculator techniques to determine factors of a polynomial using the Factor Theorem. They provide a problem involving the polynomial 3x^3 + 2x^2 - 32 and multiple choices for factors. The process involves inputting the polynomial into the calculator and testing each factor by setting it equal to zero and finding the corresponding value of x. The speaker demonstrates how to input values into the calculator, perform the calculations, and determine that x - 2 is a factor because the remainder is zero when the polynomial is divided by this factor.

The paragraph covers how to calculate the difference between a function evaluated at x+1 and the same function at x, using the function f(x) = 10^(x+1) as an example. The speaker advises using values greater than one for x to ensure accuracy and demonstrates the process of inputting the function into the calculator, evaluating it for x=2 and x=3, and then finding the difference. The correct multiple-choice answer is identified through this process, which is 900, corresponding to option D.

This section introduces the process of evaluating functions with two variables using a calculator. The function given is f(x, y) = 4x^3 + 3x^2y - 5xy^2 + y^3. The speaker explains how to input the function into the calculator, taking into account the two variables x and y, and then how to evaluate the function for specific values of x and y (x=1, y=2 in this case). The result obtained from the calculator is -2, which is used to match the correct multiple-choice answer.

The speaker presents a method for simplifying rational expressions involving powers of x and y, using the expression (x^4 - y^4) / (x^4 - 2x^2y^2 + y^4) as an example. They describe assigning values to x and y, inputting the expression into the calculator, and evaluating it to find a numerical result. The process involves checking multiple-choice answers to find the one that matches the calculated simplification, which in this case is identified as option B.

This paragraph discusses the simplification of an expression involving exponential powers of variables a, b, and c. The speaker suggests using the calculator to input and simplify the expression, assigning values to a, b, and c (2, 3, and 4 respectively), and then using the calculator to find the simplified result. The process includes storing intermediate results in the calculator for ease of comparison with multiple-choice answers, leading to the identification of the correct answer as option A.

The speaker explains how to solve for a variable in an exponential equation, using the equation x = 4^(n+2) - 24*4^(n-1) / (10*4^(n+1)) as an example. They demonstrate the process of simplifying the equation by assigning a value to the variable n and using the calculator to solve for x. The explanation includes the importance of trying different values to ensure the solution's accuracy and concludes with the determination that x equals one-fourth for any value of n used in the calculation.

The final paragraph focuses on simplifying an algebraic expression involving variables a and b, with the expression a^(-2) * b^3 / (a^2 * b^(-1)). The speaker shows how to input the expression into the calculator, using specific values for a and b (2 and 3, respectively), and then calculate the simplified result. The process involves verifying the result against multiple-choice answers and concludes with the identification of the correct answer as option C.

The last paragraph of the script deals with solving for a variable in an equation that combines exponential and multiplicative terms, with the equation a = 64^x * 4^y as an example. The speaker demonstrates the process of inputting the equation into the calculator, using specific values for x and y (2 and 3, respectively), and then solving for the numerical value of a. The explanation includes storing the calculated value in the calculator for comparison with multiple-choice answers, leading to the identification of the correct answer as option D.

In the concluding paragraph, the speaker thanks the audience for listening and encourages them to share the video with friends who are preparing for board exams. They emphasize the importance of mastering calculator techniques for success in the exams and wish the audience well, inviting them to continue learning and reviewing these techniques.