IB Maths AA · Topic 5 · 32 cards

Calculus flashcards

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What notations are used for the first and second derivative?
First derivative: \(f'(x)\) or \(\frac{dy}{dx}\). Second derivative (the derivative of the derivative): \(f''(x)\) or \(\frac{d^2y}{dx^2}\), which measures how the gradient changes.
How is the derivative defined from first principles, and what does it represent?
\(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\). It is the gradient of the tangent to the curve at \(x\), and the instantaneous rate of change of \(f\).
What is the power rule for differentiation?
If \(f(x)=ax^n\) then \(f'(x)=anx^{n-1}\), for any integer \(n\). Differentiate sums term by term; constants differentiate to 0.
State the derivatives of \(\sin x\), \(\cos x\), \(e^x\) and \(\ln x\).
\(\frac{d}{dx}\sin x=\cos x\); \(\frac{d}{dx}\cos x=-\sin x\); \(\frac{d}{dx}e^x=e^x\); \(\frac{d}{dx}\ln x=\frac1x\) (for \(x>0\)). These angle results need \(x\) in radians.
State the chain rule.
If \(y=f(g(x))\) then \(\frac{dy}{dx}=f'(g(x))\times g'(x)\). Equivalently \(\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}\): differentiate the outside, then multiply by the derivative of the inside.
State the product rule and the quotient rule.
Product: \((uv)'=u'v+uv'\). Quotient: \(\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}\).
Calculation: differentiate \(y=(x^2+1)^5\).
Chain rule: \(\frac{dy}{dx}=5(x^2+1)^4\times2x=10x(x^2+1)^4\).
How do you find the equation of a tangent and a normal to a curve at \(x=a\)?
Find \(f(a)\) and the gradient \(m=f'(a)\). Tangent: \(y-f(a)=m(x-a)\). Normal has gradient \(-\frac1m\): \(y-f(a)=-\frac1m(x-a)\).
Calculation: find the tangent to \(y=x^2\) at \(x=3\).
\(y=9\) and \(\frac{dy}{dx}=2x=6\) at \(x=3\). So \(y-9=6(x-3)\), giving \(y=6x-9\).
How do \(f'(x)\) and its sign show where a function is increasing or decreasing?
\(f'(x)>0\): \(f\) is increasing. \(f'(x)<0\): \(f\) is decreasing. \(f'(x)=0\) at a stationary point, where the tangent is horizontal.
How do you find and classify stationary points?
Solve \(f'(x)=0\). Then use the sign change of \(f'\) (positive to negative: maximum; negative to positive: minimum) or the second derivative: \(f''>0\) minimum, \(f''<0\) maximum.
What is a point of inflection, and how do you find one?
A point where the concavity changes. Solve \(f''(x)=0\) and check that \(f''\) changes sign. \(f''=0\) alone is not enough (e.g. \(x^4\) at 0).
How do you solve an optimisation problem with calculus?
Write the quantity to optimise as a function of one variable (using any constraint). Differentiate, set \(f'=0\), solve, and confirm maximum or minimum. Check the answer is in the allowed domain.
How are displacement, velocity and acceleration related by calculus?
\(v=\frac{ds}{dt}\) and \(a=\frac{dv}{dt}=\frac{d^2s}{dt^2}\). Conversely \(v=\int a\,dt\) and \(s=\int v\,dt\). Speed is \(|v|\).
What is the rule for integrating \(x^n\)?
\(\int x^n\,dx=\frac{x^{n+1}}{n+1}+c\) for \(n\neq-1\). Integrate term by term and always include the constant \(c\) for indefinite integrals.
State the integrals of \(e^x\), \(\frac1x\), \(\sin x\) and \(\cos x\).
\(\int e^x\,dx=e^x+c\); \(\int\frac1x\,dx=\ln|x|+c\); \(\int\sin x\,dx=-\cos x+c\); \(\int\cos x\,dx=\sin x+c\).
How do you integrate a composite of the form \(f(ax+b)\)?
Reverse the chain rule: integrate as normal, then divide by \(a\). For example \(\int(2x+1)^3dx=\frac{(2x+1)^4}{8}+c\).
How can you find the constant of integration \(c\)?
Use a known point on the curve or a given condition: substitute the values into the indefinite integral and solve for \(c\).
What does \(\int_a^b f(x)\,dx\) represent, and how is it evaluated?
The signed area between \(f\) and the \(x\)-axis from \(a\) to \(b\). If \(F'=f\), it equals \(F(b)-F(a)\). Areas below the axis count as negative.
Calculation: evaluate \(\int_0^2 3x^2\,dx\).
\(\left[x^3\right]_0^2=8-0=8\).
How do you find the area between two curves \(y=f(x)\) and \(y=g(x)\)?
Find the intersections to get limits \(a\) and \(b\). Then area \(=\int_a^b\big(f(x)-g(x)\big)dx\), where \(f\) is the upper curve.
What is the difference between displacement and total distance travelled when given \(v(t)\)?
Displacement \(=\int_{t_1}^{t_2}v\,dt\) (signed). Total distance \(=\int_{t_1}^{t_2}|v|\,dt\), which counts movement in both directions as positive.
(HL) What is L'Hôpital's rule?
If \(\frac{f(x)}{g(x)}\) gives \(\frac00\) or \(\frac\infty\infty\) as \(x\to a\), then \(\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}\), provided this second limit exists.
(HL) State the derivatives of \(\tan x\), \(a^x\), \(\arcsin x\) and \(\arctan x\).
\(\tan x\to\sec^2x\); \(a^x\to a^x\ln a\); \(\arcsin x\to\frac{1}{\sqrt{1-x^2}}\); \(\arctan x\to\frac1{1+x^2}\).
(HL) How does implicit differentiation work? Example: \(x^2+y^2=25\).
Differentiate every term with respect to \(x\), treating \(y\) as a function of \(x\) (so \(y^2\to2y\frac{dy}{dx}\)). Here \(2x+2y\frac{dy}{dx}=0\), so \(\frac{dy}{dx}=-\frac xy\).
(HL) How do you solve a related rates problem?
Write an equation linking the quantities, differentiate with respect to time \(t\) using the chain rule, substitute the known values and rates, then solve for the unknown rate.
(HL) What are the Maclaurin series of \(e^x\), \(\sin x\) and \(\cos x\)?
\(e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots\); \(\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots\); \(\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots\).
(HL) What is the formula for integration by parts?
\(\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dx\). Choose \(u\) as the part that simplifies when differentiated (e.g. \(x\) or \(\ln x\)).
(HL) How does integration by substitution work, including limits?
Choose \(u=g(x)\), replace \(dx\) using \(du=g'(x)dx\), and integrate in \(u\). For a definite integral, either change the limits to \(u\)-values or return to \(x\) before substituting.
(HL) What are the standard integrals leading to \(\arctan\) and \(\arcsin\)?
\(\int\frac{1}{a^2+x^2}dx=\frac1a\arctan\frac xa+c\) and \(\int\frac{1}{\sqrt{a^2-x^2}}dx=\arcsin\frac xa+c\).
(HL) How do you solve a separable differential equation \(\frac{dy}{dx}=f(x)g(y)\)?
Separate the variables: \(\int\frac{1}{g(y)}dy=\int f(x)\,dx\). Integrate both sides, include one constant, and use an initial condition to find it.
(HL) What is the volume of revolution formula for rotating \(y=f(x)\) about the \(x\)-axis?
\(V=\pi\int_a^b y^2\,dx\) for \(a\le x\le b\). About the \(y\)-axis: \(V=\pi\int_c^d x^2\,dy\). For example \(y=x\) from 0 to 3 gives \(9\pi\).

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