First Year (FE)
- How do I expand a function as a Taylor or Maclaurin series in powers of x or (x − a)? In one line Taylor's series writes f(x) as f(a)+(x-a)f'(a)+(x-a)²/2!f''(a)+dots ; Maclaurin's series is the special case a=0 . Explained sim... 1
- What are maxima and minima of a function of two variables and how do I find stationary points? In one line A function f(x,y) has a maximum (minimum) at a point if its value there is larger (smaller) than at all nearby points; such poin... 1
- How do I use the second derivative test with r, s, t to classify maxima, minima and saddle points? In one line At a stationary point compute r=f_xx , s=f_xy , t=f_yy : if rt-s²>0 it is a maximum when r<0 and a minimum when r>0 ; if rt-s²<0... 1
- What is a saddle point of a function of two variables and how do I identify it? In one line A saddle point is a stationary point that is a maximum along one direction and a minimum along another, so it is neither a maxim... 1
- How do I solve a word problem on maxima and minima, like the box with least surface area? In one line Write the quantity to optimise as a function, use the given condition to reduce it to two variables, then find and classify the... 1
- What are the nth derivatives of sin(ax+b) and cos(ax+b), and why does nπ/2 appear? In one line Dⁿsin(ax+b)=aⁿsin(ax+b+dfracnπ2) and Dⁿcos(ax+b)=aⁿcos(ax+b+dfracnπ2) : each derivative multiplies by a and shifts the angle by... 1
- How do I find the nth derivative of e^(ax) sin(bx+c) and e^(ax) cos(bx+c)? In one line Dⁿ[e^axsin(bx+c)]=rⁿe^axsin(bx+c+nφ) with r=√(a²+b²) , φ=tan⁻¹b/a ; each derivative multiplies by r and adds the angle φ . Expla... 1
- How do I find the nth derivative of a rational function using partial fractions? In one line Break the rational function into simple partial fractions like A/x-a , then apply Dⁿdfrac1ax+b=dfrac(-1)ⁿn!aⁿ(ax+b)ⁿ⁺¹ to each p... 1
- What is successive differentiation and what are the nth derivatives of e^(ax), a^(mx) and (ax+b)^m? In one line Successive differentiation means differentiating a function again and again; for standard functions we spot the pattern and writ... 1
- What is Leibnitz's theorem for the nth derivative of a product and how do I apply it? In one line Leibnitz's theorem gives the n th derivative of a product: (uv)_n=Σ_(r=0)ⁿⁿC_r u_(n-r)v_r , which looks exactly like the binomia... 1
- What are the nth derivatives of 1/(ax+b) and log(ax+b), and how are they derived? In one line Dⁿdfrac1ax+b=dfrac(-1)ⁿn! aⁿ(ax+b)ⁿ⁺¹ and Dⁿlog(ax+b)=dfrac(-1)ⁿ⁻¹(n-1)! aⁿ(ax+b)ⁿ ; signs alternate and factorials build up wit... 1
- How do I use Leibnitz's theorem to prove relations like (1−x²)y_(n+2) − (2n+1)x y_(n+1) − (n²+m²)y_n = 0? In one line First get a simple differential equation connecting y,y₁,y₂ , then differentiate it n times using Leibnitz's theorem on each pro... 1
- How do I find a power of any complex number, like (1 + i√3)⁶, using De Moivre's theorem? In one line Convert the number to polar form r(cosθ+isinθ) ; then its n th power is simply rⁿ(cos nθ+isin nθ) . Explained simply Multiplying... 1
- How do I find all the nth roots of a complex number and why are there exactly n of them? In one line Write the number as r[cos(θ+2kπ)+isin(θ+2kπ)] and apply De Moivre's theorem with power 1/n ; taking k=0,1,dots,n-1 gives the n d... 1
- How do I solve equations like x⁴ + 1 = 0 using roots of unity? In one line Rewrite the equation as xⁿ= a complex number, express that number in polar form with 2kπ added, and take the n th roots. Explain... 1
- How are sin z and cos z defined when z is a complex number, and why can they exceed 1? In one line For complex z , sin z=dfrace^iz-e^(-iz)2i and cos z=dfrace^iz+e^(-iz)2 ; these come from Euler's formula and, unlike the real ca... 1
- What are hyperbolic functions like sinh x, cosh x and tanh x, and how are they defined? In one line Hyperbolic functions are built from e^x : sinh x=dfrace^x-e^(-x)2 , cosh x=dfrace^x+e^(-x)2 , tanh x=sinh x/cosh x ; they behave... 1
- How are circular and hyperbolic functions related, for example why is sin(ix) = i sinh x? In one line Putting an imaginary argument into a circular function gives a hyperbolic function and vice versa: sin ix=isinh x , cos ix=cosh... 1
- How do I separate sin(x+iy), cos(x+iy), sinh(x+iy) and cosh(x+iy) into real and imaginary parts? In one line Expand with the addition formula and then use cos iy=cosh y , sin iy=isinh y ; for example sin(x+iy)=sin xcosh y+icos xsinh y .... 1
- What are the main hyperbolic identities and how does Osborn's rule help me remember them? In one line Every trigonometric identity has a hyperbolic twin, obtained by changing cos→cosh , sin→sinh and flipping the sign of every term... 1
- What is the logarithm of a complex number and what are its general and principal values? In one line For z=re^iθ , log z=log r+i(θ+2nπ) ; the principal value takes n=0 and θ as the principal argument. Explained simply For positiv... 1
- How do I separate tan(x+iy) into real and imaginary parts and solve problems based on it? In one line Multiply numerator and denominator by the conjugate cos(x-iy) to get tan(x+iy)=sin2x+isinh2y/cos2x+cosh2y . Explained simply For... 1
- How do I find real and imaginary parts of complex powers like i^i and (1+i)^(1−i)? In one line Use a^b=e^(blog a) : find log a for the complex base, multiply by the exponent, separate e^(u+iv)=e^u(cos v+isin v) . Explained... 1
- How do I express inverse hyperbolic functions like sinh⁻¹x, cosh⁻¹x and tanh⁻¹x as logarithms? In one line Because hyperbolic functions are built from e^x , their inverses are logarithms: sinh⁻¹x=log(x+√(x²+1)) , cosh⁻¹x=log(x+√(x²-1))... 1