# Curve Sketching #1 - Weebly CURVE SKETCHING #1 AP CALCULUS NOVEMBER 9-10, 2016 MRS. AGNEW ESSENTIAL QUESTION WHAT IS THE SIGNIFICANCE OF THE MEAN VALUE THEOREM? HOW DO YOU SKETCH CURVES USING DERIVATIVES?

ESSENTIAL VOCABULARY MEAN VALUE THEOREM FIRST DERIVATIVE TEST INCREASING/DECREASING FUNCTION ZEROS & SYMMETRY ZEROS OF A FUNCTION WHERE GRAPH INTERSECTS XAXIS

SYMMETRY Y-AXIS (PLUG IN X FOR X) ORIGIN (PLUG IN X AND Y) DISCONTINUITIE S ASYMPTOTES VERTICAL: WHERE FUNCTION IS UNDEFINED (NOT HOLE)

HORIZONTAL: DEGREE OF NUMERATOR VS. DENOMINATOR SLANT: DEGREE OF NUMERATOR IS ONE MORE THAN DENOMINATOR POINT OF DISCONTINUITY (HOLE) Examples INCREASING/ DECREASING WHAT ARE CRITICAL NUMBERS?

IF F (X) > 0 ON AN INTERVAL, THEN F(X) IS INCREASING ON THAT INTERVAL. IF F (X) < 0 ON AN INTERVAL, THEN F(X) IS DECREASING ON THAT INTERVAL. FIRST DERIVATIVE TEST a)

GIVEN C IS A CRITICAL NUMBER OF F(X) IF F CHANGES FROM + TO AT C, THEN F HAS A RELATIVE MAXIMUM AT C. b) IF F CHANGES FROM TO + AT C, THEN F HAS A RELATIVE MINIMUM AT C. c) IF F DOES NOT CHANGE AT C, THEN F HAS NO EXTREME VALUE. Examples

MEAN VALUE IF F IS CONTINUOUS OVER [A,B] AND THEOREM DIFFERENTIABLE OVER (A,B), THEN THERE EXISTS A NUMBER C BETWEEN A AND B SUCH THAT CALCULUS SLOPE = ALGEBRA SLOPE INSTANTANEOUS ROC = AVERAGE ROC

Animation ROLLES THEOREM LET F BE CONTINUOUS ON [A,B] AND DIFFERENTIABLE OVER (A,B). IF F(A) = F(B), THEN THERE EXIST AT LEAST ONE C SUCH F (C) = 0. GUARANTEES THE EXISTENCE OF AN EXTREME VALUE IN THE INTERIOR OF A CLOSED INTERVAL.

Examples HOMEWORK: PAGE 176 177 #13, 17, 19, 21, 29, 41 46, 53, 64, 71 PAGE 186 189 #19, 23, 29, 35, 49, 67, 69, 85, 91