4.3 CURVE SKETCING CURVE SKETCHING To draw a curve without a calculator, we must know the following things: 1. Zeros 2. 3. 4. Asymptotes Points of discontinuity Local mins/local maxes 5. Set orginal function = 0 and solve Find critical points by taking derivative, setting = 0, and solving. Concavity FIRST DERIVATIVE TEST We have already discussed how to determine if a point is a local min or a local max. First Derivative Test At a critical point, if f’ changes signs, then it is a local extreme. If f’ does not change sign at a critical point, then this point is not an extreme value. At endpoints: If f’ < 0 after the left endpoint, then the left endpoint is a local max. If f’ > 0 after the left endpoint, then the left endpoint is a local min. If f’ < 0 before the right endpoint, then the right endpoint is a local min. If f’ > 0 before the right endpoint, then the right endpoint is a local max. FIRST DERIVATIVE TEST Example: Use the First Derivative Test to find the local extreme values of g(x) = (x2 – 3)ex. CONCAVITY Concavity talks about the way a graph is turned. “Smiley face” = concave up “Frowny face” = concave down To find where a graph is concave up, find the points of inflection The place on a graph where concavity changes. Find second derivative and set = 0. CONCAVITY Example: Find all points of inflection of the graph of y = e-x2. CURVE SKETCHING Example Sketch the curve f(x) = x3 – 5x2 + 3x + 6. CURVE SKETCHING Example: A particle is moving along the x-axis with position function x(t) = 2t3 – 14t2 + 22t – 5 for t ≥ 0. Find the velocity and acceleration and describe the motion of the particle. SECOND DERIVATIVE TEST Another way to determine local mins/local maxes is by the Second Derivative Test. If at a critical point f’’(c) is negative (concave down), then f has a local max at x = c. If at a critical point f’’(c) is positive (concave up), then f has a local min at x = c. Note: You cannot use this test is f’’(c) = 0 or f’’(c) fails to exist.