![]() ![]() Determine x- and y-intercepts and vertical and horizontal asymptotes when appropriate. State clearly the intervals on which the function is increasing ( ) Determine all relative and absolute maximum and minimum values and inflection points. These are the directions for problems 1 through 10. Ī.) If f''( c)0 then f has a relative minimum value at x= c. and 6.) : Assume that y= f( x) is a twice-differentiable function with f'( c)=0. THE SECOND DERIVATIVE TEST FOR EXTREMA (This can be used in place of statements 5. The point x= a determines an inflection point for function f if f is continuous at x= a, and the second derivative f'' is negative (-) for x a, or if f'' is positive (+) for x a. The point x= a determines an absolute minimum for function f if it corresponds to the smallest y-value in the range of f. The point x= a determines a relative minimum for function f if f is continuous at x= a, and the first derivative f' is negative (-) for x a. The point x= a determines an absolute maximum for function f if it corresponds to the largest y-value in the range of f. The point x= a determines a relative maximum for function f if f is continuous at x= a, and the first derivative f' is positive (+) for x a. ![]() If the second derivative f'' is negative (-), then the function f is concave down ( )ĥ. If the second derivative f'' is positive (+), then the function f is concave up ( )Ĥ. If the first derivative f' is negative (-), then the function f is decreasing (ģ. If the first derivative f' is positive (+), then the function f is increasing ( )Ģ. To avoid overlooking zeroes in the denominators of f' and f'', it is helpful to rewrite all negative exponents as positive exponents and then carefully manipulate and simplify the resulting fractions.ġ. Establish a sign chart (number line) for f'' in the exact same manner. ( x, y) will be a starting point for the graph of f. Beneath each designated x-value, write the corresponding y-value which is found by using the original equation y = f( x). Next, pick points between these designated x-values and substitute them into the equation for f' to determine the sign ( + or - ) for each of these intervals. These designated x-values establish intervals along the sign chart. Above these x-values and the sign chart draw a dotted vertical line to indicate that the value of f' does not exist at this point. For example, mark those x-values where division by zero occurs in f'. In addition, mark x-values where the derivative does not exist (is not defined). Mark these x-values underneath the sign chart, and write a zero above each of these x-values on the sign chart. To establish a sign chart (number lines) for f', first set f' equal to zero and then solve for x. Here are instruction for establishing sign charts (number line) for the first and second derivatives. In addition, it is important to label the distinct sign charts for the first and second derivatives in order to avoid unnecessary confusion of the following well-known facts and definitions. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by carefully labeling critical points, intercepts, and inflection points. ![]() Problems range in difficulty from average to challenging. The following problems illustrate detailed graphing of functions of one variable using the first and second derivatives. GRAPHING OF FUNCTIONS USING FIRST AND SECOND DERIVATIVES ![]() Graphing Using First and Second Derivatives ![]()
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |