IB Maths AI · Topic 5 · 32 cards

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What does the derivative \(f'(x)\) or \(\dfrac{dy}{dx}\) represent?
The gradient of the tangent to the curve at \(x\), i.e. the instantaneous rate of change of \(y\) with respect to \(x\).
How is the derivative defined using a limit?
\(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}h\): the gradient of a chord whose two points move together.
What is the power rule? Differentiate \(3x^4-5x+2\).
\(\dfrac d{dx}(ax^n)=anx^{n-1}\); constants differentiate to 0. So \(f'(x)=12x^3-5\).
How do you differentiate \(\dfrac4{x^2}\)?
Rewrite as \(4x^{-2}\) first, then use the power rule: \(-8x^{-3}=-\dfrac8{x^3}\). Rewrite reciprocals as powers before differentiating.
What is the difference between average and instantaneous rate of change?
Average: the gradient of a chord over an interval \(\frac{\Delta y}{\Delta x}\). Instantaneous: the gradient of the tangent at one point, given by the derivative.
How do you find the gradient of a curve at a point? Find it for \(f(x)=x^2-4x\) at \(x=3\).
Differentiate, then substitute: \(f'(x)=2x-4\), so \(f'(3)=2\).
How do you find the equation of a tangent? Find the tangent to \(y=x^2\) at \(x=2\).
Use \(y-f(a)=f'(a)(x-a)\). Here \(f(2)=4\) and \(f'(2)=4\): \(y-4=4(x-2)\), i.e. \(y=4x-4\).
How do you find the equation of a normal? What is the normal gradient for \(y=x^2\) at \(x=2\)?
The normal is perpendicular to the tangent, so its gradient is \(-\dfrac1{f'(a)}\). Here \(-\tfrac14\). Then use the point-gradient form.
How do you tell where a function is increasing or decreasing?
Increasing where \(f'(x)>0\) and decreasing where \(f'(x)<0\). Find the zeros of \(f'\), then test the sign on each side.
How do you find and classify a stationary point using the first derivative?
Solve \(f'(x)=0\). If \(f'\) changes from + to − it is a local maximum; − to + a local minimum; no change means neither.
How can you find the maximum or minimum point of a function with the GDC?
Graph the function and use the max or min tool within a chosen window, or the numerical derivative to find where the gradient is 0. Give coordinates to 3 s.f.
What are the steps of an optimisation problem? Find the maximum of the area \(A=x(10-x)\).
Form a function of one variable, differentiate, set \(=0\), check it is a max or min, then answer in context. \(A'=10-2x=0\), so \(x=5\) and \(A=25\).
How do you check whether a local maximum is also the greatest value on a domain?
Compare the value at the stationary point with the values at the endpoints of the domain. The largest of these is the global maximum.
How do you find a function's graph shape from its derivative graph?
Zeros of \(f'\) are the x-positions of the stationary points of \(f\). Where \(f'\) is above the axis \(f\) is increasing; where below, decreasing.
What is the unit of a derivative in context?
Units of \(y\) per unit of \(x\). If \(V\) is in litres and \(t\) in minutes, then \(\dfrac{dV}{dt}\) is in litres per minute.
What is the rule for integrating powers? Find \(\int6x^2\,dx\).
\(\int ax^n\,dx=\dfrac{a}{n+1}x^{n+1}+C\), for \(n\ne-1\). So \(\int6x^2dx=2x^3+C\).
How do you find the constant of integration? If \(f'(x)=2x\) and \(f(1)=5\), find \(f\).
Integrate: \(f=x^2+C\). Substitute the point: \(1+C=5\), so \(C=4\) and \(f(x)=x^2+4\).
What is a definite integral, and how is it evaluated? Find \(\int_0^3x^2\,dx\).
The signed area between the curve and the x-axis from \(a\) to \(b\): \(F(b)-F(a)\). Here \(\dfrac{3^3}3-0=9\).
How do you find the area between a curve and the x-axis when the curve crosses the axis?
Split the integral at each x-intercept and add the absolute values (or integrate \(|f(x)|\) on the GDC); otherwise areas below the axis cancel with those above.
How do you find the area between two curves?
Find the intersections, then \(\int_a^b\big(f(x)-g(x)\big)dx\) where \(f\) is the upper curve. Use the GDC to find the limits and the integral.
What is the trapezoidal rule? Estimate the area for \(y=1,3,5\) at equal spacing \(h=2\).
\(A\approx\dfrac h2\big[y_0+2(y_1+\dots+y_{n-1})+y_n\big]\). Here \(\dfrac22(1+6+5)=12\).
(HL) What is the chain rule? Differentiate \((3x+1)^5\).
\(\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}\): differentiate the outer function, then multiply by the derivative of the inner. So \(5(3x+1)^4\times3=15(3x+1)^4\).
(HL) What are the product and quotient rules?
Product: \((uv)'=u'v+uv'\). Quotient: \(\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}\). For example, \((x\sin x)'=\sin x+x\cos x\).
(HL) What are the derivatives of \(e^x\), \(\ln x\), \(\sin x\) and \(\cos x\)?
\((e^x)'=e^x\); \((\ln x)'=\dfrac1x\); \((\sin x)'=\cos x\); \((\cos x)'=-\sin x\) (angles in radians). With a linear inside, e.g. \((e^{2x})'=2e^{2x}\).
(HL) How does the second derivative describe concavity and inflection points?
\(f''>0\): concave up; \(f''<0\): concave down. A point of inflection is where \(f''\) changes sign (usually \(f''=0\)). A stationary point with \(f''>0\) is a minimum, with \(f''<0\) a maximum.
(HL) How are displacement, velocity and acceleration linked using calculus?
\(v=\dfrac{ds}{dt}\) and \(a=\dfrac{dv}{dt}\); conversely \(v=\int a\,dt\) and \(s=\int v\,dt\). Constants are found from initial conditions.
(HL) How do you find total distance rather than displacement from a velocity function?
Displacement is \(\int_{t_1}^{t_2}v\,dt\). Distance is \(\int_{t_1}^{t_2}|v|\,dt\): split where \(v=0\) and add magnitudes.
(HL) What are the standard integrals of \(e^x\), \(\dfrac1x\), \(\sin x\) and \(\cos x\)?
\(\int e^x dx=e^x+C\); \(\int\dfrac1x dx=\ln|x|+C\); \(\int\sin x\,dx=-\cos x+C\); \(\int\cos x\,dx=\sin x+C\).
(HL) How does integration by substitution work? Find \(\int2x(x^2+1)^3dx\).
Let \(u=x^2+1\), so \(du=2x\,dx\): \(\int u^3du=\dfrac{u^4}4+C=\dfrac{(x^2+1)^4}4+C\).
(HL) How do you find a volume of revolution about the x-axis? Find it for \(y=x\) from 0 to 3.
\(V=\pi\int_a^by^2\,dx\). Here \(\pi\int_0^3x^2dx=9\pi\), the volume of a cone of radius 3 and height 3.
(HL) How do you solve a separable differential equation \(\dfrac{dy}{dx}=\dfrac{g(x)}{h(y)}\)?
Separate variables: \(\int h(y)\,dy=\int g(x)\,dx\). Integrate both sides, include one constant, and use the initial condition to find it.
(HL) How does Euler's method approximate a solution of \(\dfrac{dy}{dx}=f(x,y)\)?
Step by step: \(x_{n+1}=x_n+h\), \(y_{n+1}=y_n+h\,f(x_n,y_n)\). Smaller \(h\) gives a better approximation. For \(y'=x+y\), \(y(0)=1\), \(h=0.1\): \(y_1=1.1\).

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