What you’ll learn11 learning objectivesChoose one objective for a focused lesson, or study the complete topic.—SL 5.1—Limits and derivative concept• Estimate limits from tables or graphs; formal analytic limit methods are not required.• Interpret derivative as gradient function and rate of change.• Use notation dy/dx, f'(x), dV/dr and ds/dt.Syllabus objective—SL 5.2—Increasing and decreasing functions• Interpret f'(x)>0, f'(x)=0 and f'(x)<0 graphically.• Identify intervals where functions are increasing or decreasing.Syllabus objective—SL 5.3—Derivative of powers• Differentiate f(x)=ax^n to f'(x)=anx^(n-1), n in Z.• Differentiate sums of integer-power terms.Syllabus objective—SL 5.4—Tangents and normals• Find tangents and normals at a given point and their equations.• Use analytic methods and technology.Syllabus objective—SL 5.5—Integration as anti-differentiation• Integrate polynomial-type functions as anti-derivatives.• Use boundary conditions to determine constants.• Connect anti-derivatives, definite integrals and area under curves.Syllabus objective—SL 5.6—Differentiation rules• Differentiate x^n (n in Q), sin x, cos x, e^x and ln x.• Use sum/multiple rules, chain rule, product rule and quotient rule.Syllabus objective—SL 5.7—Second derivative and graph behaviour• Use second derivative notation d2y/dx2 and f''(x).• Relate graphs of f, f' and f''; use technology where useful.Syllabus objective—SL 5.8—Extrema, optimization and inflexion• Find local maxima/minima using first-derivative sign changes or second derivative tests.• Solve optimization problems in contexts such as profit, area and volume.• Identify points of inflexion and concavity.Syllabus objective—SL 5.9—Kinematics• Use displacement s, velocity v, acceleration a and total distance travelled.• v=ds/dt and a=dv/dt=d2s/dt2.• Displacement is integral of velocity; distance is integral of speed |v(t)|.Syllabus objective—SL 5.10—Indefinite integration• Integrate x^n, sin x, cos x, 1/x and e^x.• Use composites with linear functions and reverse chain rule/substitution by inspection.Syllabus objective—SL 5.11—Definite integrals and areas• Use definite integral analytically: integral_a^b g'(x) dx = g(b)-g(a).• Find areas under curves and between curves, accounting for sign.• Write correct integral expressions before calculating.Syllabus objective