FortyFive › Revision notes › IB Maths: Analysis & Approaches › IB Maths: Analysis & Approaches: Calculus
Limits, differentiation, integration, kinematics, optimisation. 19 subtopics.
In this topic: 5.1 · 5.2 · 5.3 · 5.4 · 5.5 · 5.6 · 5.7 · 5.8 · 5.9 · 5.10 · 5.11 · 5.12 · 5.13 · 5.14 · 5.15 · 5.16 · 5.17 · 5.18 · 5.19
The derivative is the gradient of the tangent at a point and the instantaneous rate of change.
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\(f'(x)\gt0\): \(f\) is increasing. \(f'(x)\lt0\): \(f\) is decreasing. \(f'(x)=0\): stationary point.
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Rewrite fractions and roots as powers first: \(\dfrac{5}{x^{2}}=5x^{-2}\), \(\sqrt x=x^{\frac12}\).
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\text{Tangent at } x=a:\quad y-f(a)=f'(a)(x-a)\text{Normal gradient}=-\frac{1}{f'(a)}
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Integration reverses differentiation. Use a boundary condition to find \(C\).
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\frac{d}{dx}\sin x=\cos x\qquad\frac{d}{dx}\cos x=-\sin x\qquad\frac{d}{dx}e^{x}=e^{x}\qquad\frac{d}{dx}\ln x=\frac1x\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}\qquad (uv)'=u'v+uv'\qquad\left(\frac uv\right)'=\frac{u'
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\(f''(x)\gt0\): concave up. \(f''(x)\lt0\): concave down.
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Stationary points: solve \(f'(x)=0\), then test with \(f''\) or a sign table.
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The particle changes direction where \(v=0\) and \(v\) changes sign.
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Integrate by inspection or substitution when you spot a function multiplied by (a multiple of) its derivative.
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If the curve goes below the \(x\)-axis, integrate each section separately and add the absolute values.
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f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\qquad f^{(n)}(x)=\frac{d^{n}y}{dx^{n}}
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Apply repeatedly if the result is still indeterminate.
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Differentiate every term with respect to \(x\), then make \(\frac{dy}{dx}\) the subject.
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Reverse these for integrals, and use partial fractions for rational functions.
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Choose \(u\) to become simpler when differentiated (logarithms first, then polynomials).
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\text{Area with the }y\text{-axis}=\int_a^b x\,dyV=\pi\int_a^b y^{2}\,dx\ \ (\text{about the }x\text{-axis})\qquad V=\pi\int_a^b x^{2}\,dy\ \ (\text{about the }y\text{-axis})
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\text{Euler: }y_{n+1}=y_n+h\,f(x_n,y_n),\quad x_{n+1}=x_n+h\text{Separable: }\int g(y)\,dy=\int f(x)\,dx\text{Homogeneous }\tfrac{dy}{dx}=f\!\left(\tfrac yx\right):\ \text{substitute }y=vx\text{Linear }\tfrac{dy}{dx}+P(x
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Build new series by substitution, multiplication, differentiation or integration of known series.
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