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MH HSC 12 Maths

Application of Derivatives

Each concept is a short lesson, then questions until you get three right.

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10 concepts, a short lesson each

  • Tangents and rates0/5
  • Mean value theorems0/2
  • Increasing functions and extremes0/3

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Tangent and normal at a point

The tangent at (x1,y1)(x_1, y_1) has slope m=(dydx)(x1,y1)m = \left(\dfrac{dy}{dx}\right)_{(x_1, y_1)}: y−y1=m(x−x1)y - y_1 = m(x - x_1); the normal has slope −1m-\dfrac{1}{m}

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Tangents and rates0 of 5

  1. Tangent and normal at a pointThe tangent at (x1,y1)(x_1, y_1) has slope m=(dydx)(x1,y1)m = \left(\dfrac{dy}{dx}\right)_{(x_1, y_1)}: y−y1=m(x−x1)y - y_1 = m(x - x_1); the normal has slope −1m-\dfrac{1}{m}Start
  2. Tangents with a given slopeSet dydx\dfrac{dy}{dx} equal to the slope wanted (parallel: equal; perpendicular: −1m-\dfrac{1}{m}) and solve for the points of contactStart
  3. Derivative as a rate of changedydx\dfrac{dy}{dx} is the rate at which yy changes with xx; for motion s(t)s(t), velocity v=dsdtv = \dfrac{ds}{dt} and acceleration a=dvdta = \dfrac{dv}{dt}Start
  4. Related ratesWrite the relation between the quantities, differentiate both sides with respect to tt, then substitute the known rates and valuesStart
  5. ApproximationsNear aa, the curve hugs its tangent: f(a+h)≈f(a)+h f′(a)f(a + h) \approx f(a) + h\,f'(a), as in 8.95≈3−0.056\sqrt{8.95} \approx 3 - \dfrac{0.05}{6}Start

Mean value theorems0 of 2

  1. Rolle's theoremIf ff is continuous on [a,b][a, b], differentiable on (a,b)(a, b) and f(a)=f(b)f(a) = f(b), then f′(c)=0f'(c) = 0 for some cc in (a,b)(a, b): a horizontal tangentStart
  2. Lagrange's mean value theoremIf ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), some cc in (a,b)(a, b) has f′(c)=f(b)−f(a)b−af'(c) = \dfrac{f(b) - f(a)}{b - a}: a tangent parallel to the chordStart

Increasing functions and extremes0 of 3

  1. Increasing and decreasing functionsff increases where f′(x)>0f'(x) > 0 and decreases where f′(x)<0f'(x) < 0: find where f′=0f' = 0 and test the sign of f′f' between those pointsStart
  2. Local maxima and minimaAt a critical point f′(c)=0f'(c) = 0: a maximum if f′′(c)<0f''(c) < 0, a minimum if f′′(c)>0f''(c) > 0; or watch f′f' change sign from ++ to −- (maximum) or −- to ++Start
  3. Greatest and least values in problemsWrite the quantity to optimise in one variable using the constraint, find its critical point, and check it is the maximum or minimum asked forStart

Practise Application of Derivatives

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions