Thus the derivative of the ratio is proportional to that same ratio and the

# Thus the derivative of the ratio is proportional to

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5.2 Higher derivatives and antiderivatives Section 5.2 Learning goals 1. 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.
90 DIFFERENTIAL CALCULUS FOR LIFE SCIENCES Example 5.6.Compute the following derivatives(a) Find y000if y=x5-4x3+10x2-3,(b) Findd2ydt2if y=12t+4t,(c) Find D2f for the function f(t) =At3+Bt2+Ct+D,(d) Find f000(x)if f(x) =8x. 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.
DIFFERENTIATION 91 t y y t Figure 5.1: Function graphs for Exam- ple 5.7 ; one smooth, the other with cusps. Solution. In Figure 5.2 we show the functions y ( t ) (top row), their first derivatives y 0 ( t ) (middle row), and the second derivatives y 00 ( t ) (bottom row). In each case, we determined the slopes of tangent lines as a first step. 0 + + + + 0 t y y t 0 + 0 - - 0 y t 0 + 0 y t y t 0 0 0 y t ( a ) ( b ) Figure 5.2: Sketching the solutions for Example 5.7 : 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), the derivative is zero. This is indicated at several places in Figure 5.2 . In (b), there are several cusps at which the first and second derivatives are not defined. Mastered Material Check
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