♦5.2Higher derivatives and antiderivativesSection5.2Learning goals1. Given a sketch of a function, sketch its first and second derivatives.2. Given the sketch of a function, sketch its first and second antiderivatives.
90DIFFERENTIAL CALCULUS FOR LIFE SCIENCESExample 5.6.Compute the following derivatives(a) Find y000if y=x5-4x3+10x2-3,(b) Findd2ydt2if y=12t+4√t,(c) Find D2f for the function f(t) =At3+Bt2+Ct+D,(d) Find f000(x)if f(x) =8√x.♦In Section4.1we learned how to sketch the derivative of a function froma sketch of its graph. By repeating this operation we can now sketch secondderivatives or even higher order derivatives.Example 5.7(Sketching derivatives).Sketch the first and second derivativesthe functions given in Figure5.1.
DIFFERENTIATION91tyytFigure 5.1: Function graphs for Exam-ple5.7; one smooth, the other with cusps.Solution.In Figure5.2we show the functionsy(t)(top row), their firstderivativesy0(t)(middle row), and the second derivativesy00(t)(bottom row).In each case, we determined the slopes of tangent lines as a first step.0++++0tyyt0+0- -0yt0+0ytyt000yt(a)(b)Figure 5.2: Sketching the solutions forExample5.7: antiderivatives.Recall that wherever a tangent line to a curve is horizontal, (e.g., at the“tops of peaks” or “bottoms of valleys” or at flat parts of the graph), thederivative is zero. This is indicated at several places in Figure5.2. In (b),there are several cusps at which the first and second derivatives are notdefined.♦Mastered Material Check