 Practice Problems - Page 42      Custom Search   PRACTICE PROBLEMS: Differentiate the functions in problems 1 through 3. 4. Find the slope of the tangent line to the curve at the points x = - 2, 0, and 3 5. Find the values of x where the function has a maximum or a minimum. ANSWERS: SUMMARY The following are the major topics covered in this chapter: 2. Indeterminate forms: Two methods of evaluating indeterminate forms are (1) fac�toring and (2) division of the numerator and denominator by powers of the variable. 3. Limit theorems: Theorem 1. The limit of the sum of two functions is equal to the sum of the limits: This theorem can be extended to include any number of functions, such as Theorem 2. The limit of a constant, c, times a function, fx), is equal to the constant, c, times the limit of the function: Theorem 3. The limit of the product of two functions is equal to the product of their limits: Theorem 4. The limit of the quotient of two functions is equal to the quotient of their limits, provided the limit of the divisor is not equal to zero: 4. Infinitesimals: A variable that approaches 0 as a limit is called an infinitesimal: The difference between a variable and its limit is an infinitesimal: If lim V = L, then lim V - L = 0 5. Sum and product of infinitesimals: Theorem 1. The algebraic sum of any number of infinitesimals is an infinitesimal. Theorem 2. The product of any number of infinitesimals is an infinitesimal. Theorem 3. The product of a constant and an infinitesimal is an infinitesimal. 6. Continuity: A function, f(x), is continuous at x = a if the following three conditions are met: 7. Discontinuity: If a function is not continuous at x = a, then it is said to be discontinuous at x = a. 8. Ways of expressing the derivative of y with respect to x: 9. Increment method for obtaining the derivative of y with respect to x: 1. Set up the function of x as a function of (x + x) and expand this function. 2. Subtract the original function of x from the new function of (x + x). 3. Divide both sides of the equation by x. 4. Take the limit of all the terms in the equation as x approaches zero. The resulting equation is the derivative of f(x) with respect to x. General formula for the derivative of any expression in x: 11. Maximum or minimum points on a curve: Set the derivative of the function, f(x), equal to zero and determine the values of the independent variable that will make the derivative equal to zero. (Note: When the derivative of a function is set equal to zero, that does not mean in all cases the curve will have a maximum or minimum point.) ADDITIONAL PRACTICE PROBLEMS Find the limit of each of the following:   ANSWERS TO ADDITIONAL PRACTICE PROBLEMS  Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business