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Formulas: calculus, grade 11 (4 units)

Transformations of the graph of $f$

  • A horizontal shift: p\displaystyle p units to the right (p>0\displaystyle p>0) or to the left (p<0\displaystyle p<0)
    y=f(x−p)\displaystyle y=f(x-p)
  • A vertical shift: k\displaystyle k units up (k>0\displaystyle k>0) or down (k<0\displaystyle k<0)
    y=f(x)+k\displaystyle y=f(x)+k
  • A reflection in the x\displaystyle x-axis
    y=−f(x)\displaystyle y=-f(x)
  • A reflection in the y\displaystyle y-axis
    y=f(−x)\displaystyle y=f(-x)
  • A vertical stretch (a>1\displaystyle a>1) or squeeze (0<a<1\displaystyle 0<a<1)
    y=a⋅f(x)\displaystyle y=a\cdot f(x)
  • Absolute value: the part below the x\displaystyle x-axis is reflected above it
    y=∣f(x)∣\displaystyle y=|f(x)|

The rational function

  • The horizontal asymptote when the numerator and the denominator have the same degree – a\displaystyle a and b\displaystyle b the coefficients of the highest power
    y=ab\displaystyle y=\frac{a}{b}
  • The horizontal asymptote when the numerator’s degree is lower than the denominator’s
    y=0\displaystyle y=0
  • An even function – the graph is symmetric about the y\displaystyle y-axis
    f(−x)\displaystyle f(-x) =\displaystyle = f(x)\displaystyle f(x)
  • An odd function – the graph is symmetric about the origin
    f(−x)\displaystyle f(-x) =\displaystyle = −f(x)\displaystyle -f(x)

Derivatives and the tangent

  • The derivative of a power
    (xn)′=nxn−1\displaystyle \left(x^{n}\right)'=nx^{n-1}
  • The derivative of a product
    (u⋅v)′\displaystyle \left(u\cdot v\right)' =\displaystyle = u′⋅v+u⋅v′\displaystyle u'\cdot v+u\cdot v'
  • The derivative of a quotient
    (uv)′\displaystyle \left(\frac{u}{v}\right)' =\displaystyle = u′⋅v−u⋅v′v2\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}
  • The derivative of cx\displaystyle \frac{c}{x}
    (cx)′\displaystyle \left(\frac{c}{x}\right)' =\displaystyle = −cx2\displaystyle -\frac{c}{x^{2}}
  • The derivative of a root – u\displaystyle u a function of x\displaystyle x
    (u)′\displaystyle \left(\sqrt{u}\right)' =\displaystyle = u′2u\displaystyle \frac{u'}{2\sqrt{u}}
  • The equation of the tangent at the point (x1,y1)\displaystyle (x_1,y_1)
    y−y1\displaystyle y-y_1 =\displaystyle = m(x−x1)\displaystyle m(x-x_1)
  • The slope of the tangent
    m\displaystyle m =\displaystyle = f′(x1)\displaystyle f'(x_1)

Investigating a function with its derivative

  • The function rises
    f′(x)>0\displaystyle f'(x)>0
  • The function falls
    f′(x)<0\displaystyle f'(x)<0
  • A possible extremum point – check the sign of f′\displaystyle f' on each side of it
    f′(x)=0\displaystyle f'(x)=0

Integrals and areas

  • The integral of a power
    ∫xn dx\displaystyle \int x^{n}\,dx =\displaystyle = xn+1n+1+C\displaystyle \frac{x^{n+1}}{n+1}+Cn≠−1\displaystyle n\neq-1
  • The integral of c(ax+b)2\displaystyle \frac{c}{(ax+b)^{2}}
    ∫c(ax+b)2 dx\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx =\displaystyle = −ca(ax+b)+C\displaystyle -\frac{c}{a(ax+b)}+C
  • The definite integral – F\displaystyle F an antiderivative of f\displaystyle f
    ∫abf(x) dx\displaystyle \int_{a}^{b}f(x)\,dx =\displaystyle = F(b)−F(a)\displaystyle F(b)-F(a)
  • The area between two graphs, from x=a\displaystyle {x}={a} to x=b\displaystyle {x}={b}
    S\displaystyle S =\displaystyle = ∫ab(f(x)−g(x))dx\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dxf(x)≥g(x)\displaystyle f(x)\geq g(x)