the concert -eat and Calculus and Vectors - How to get an A+ 4.5 An Algorithm for Curve Sketching A Algorithm for Curve Sketching Ex 1. Sketch the graph for y = f(x) = 3.x - 5x3 1. Domain denominator = 0 (rational functions) radicand 2 0 (even roots) > logarithmic argument > 0 (Icgarithmic functions) 2. Intercepts f(x) =0 (x-intercepts or zeros) numerator =0 (for rational functions) y - int = f(0) (if exists) 3. Symmetry - f(-x) = f(x) (even functions are symmetric about the y-axis) - f(-x) = -f(x) (odd functions are symmetric about the origin) = f(x + I) =f(x) (periodic functions have cycles) 4. Asymptotes - compute lim f(x) (horizontal asymptote) - compute lim f'(x) (vertical asymptote where a is a zero of the denominator but not of the numerator) compute long division (to find the oblique asymptotes for rational functions) 5. First Derivative - compute f'(-x) find critical points ( f' (x) = 0 or f'(x) DNE) create the sign chart for f'(x) find intervals of increase/decrease find the local extrema (using first derivative test) and global extrema (if function is defined on a closed interval, 6. Second Derivative compute f"(x) find points where f"(x) = 0 or f"(x) DNE create the sign chart for f"(.x) find points of inflection - find intervals of concavity upward/downward check the local extrema using the second derivative test (if necessary) 7. Curve Sketching use broken lines to draw the asymptotes plot x- and y- intercepts, extrema, and inflection points draw the curve near the asymptotes - sketch the curve 4.5 An Algorithm for Curve Sketching @2010 Julia & Teodoru Gugoiu - Page 1 of 4Calculus and Vectors - How to get an A+ Ex 2. Sketch the graph for y = f(x) = x -6.x- +9x+1. Ex 3. Sketch the graph for y = f (x) = -2 4.x 7 2 + 1 4.5 An Algorithm for Curve Sketching @2010 lulia & Teodoru Gugoiu - Page 2 of 4