Ch 6 applies derivatives to real-world problems: finding rates of change, equations of tangents and normals, determining when functions increase or decrease, and finding maximum/minimum values.
dy/dx gives the rate of change of y with respect to x. The slope of the tangent to y = f(x) at x = a is f'(a). Tangent equation: y − f(a) = f'(a)(x − a). Normal is perpendicular to the tangent.
f is increasing on (a,b) if f'(x) > 0 for all x in (a,b). f is decreasing if f'(x) < 0. Critical points occur where f'(x) = 0 or f'(x) is undefined.
First derivative test: if f' changes from + to − at c, local maximum. If − to +, local minimum. Second derivative test: if f'(c)=0 and f''(c) > 0 → local minimum. f''(c) < 0 → local maximum.
Download: https://ncert.nic.in/textbook/pdf/lemh106.pdf | Complete book Part I: https://ncert.nic.in/textbook/pdf/lemh1ps.zip
Find all critical points in the interval and evaluate f at those points and at the endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.
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