IB Maths: Analysis and Approaches HL 166711

Introduction to the concept of a limit.

Estimation of the value of a limit from a table or graph.

Not required: Formal analytic methods of calculating limits.

Derivative interpreted as gradient function and as rate of change.

Forms of notation: $\dfrac{\text{d}y}{\text{d}x}$, $f'(x)$, $\dfrac{\text{d}V}{\text{d}r}$ or $\dfrac{\text{d}s}{\text{d}t}$ for the first derivative.

Informal understanding of the gradient of a curve as a limit.

Increasing and decreasing functions.

Graphical interpretation of $f'(x)>0$, $f'(x)=0$, $f'(x)<0$.

Identifying intervals on which functions are increasing ($f'(x)>0$) or decreasing ($f'(x)<0$).

Derivative of $f(x)=ax^n$ is $f'(x)=anx^{n−1}, n∈ℤ$

The derivative of functions of the form

$f(x)=ax^n+bx^{n−1} ...$

where all exponents are integers.

Tangents and normals at a given point, and their equations.

Use of both analytic approaches and technology.

Introduction to integration as anti-differentiation of functions of the form $f(x)=ax^n+bx^{n−1}+...$, where $n∈ℤ, n≠−1$

Students should be aware of the link between anti-derivatives, definite integrals and area.

Anti-differentiation with a boundary condition to determine the constant term.

Example: If $\dfrac{\text{d}y}{\text{d}x}=3x^2+x$ and $y=10$ when $x=1$,

then $y=x^3+12x^2+8.5$.

Definite integrals using technology.

Area of a region enclosed by a curve $y=f(x)$ and the $x$-axis, where $f(x)>0$.

Students are expected to first write a correct expression before calculating the area, for example $\displaystyle\int_2^6(3x^2+4) \text{d}x$.

The use of dynamic geometry or graphing software is encouraged in the development of this concept.

Derivative of $x^n, (n∈ℚ)$, $\sin{x}$, $\cos{x}$, $e^x$ and $\ln{x}$.

Differentiation of a sum and a multiple of these functions.

The chain rule for composite functions.

Example:

• $f(x)=e^{(x^2+2)}$,
• $f(x)=\sin{(3x−1)}$

The product and quotient rules.

Link to: composite functions (SL 2.5).

The second derivative.

Graphical behaviour of functions, including the relationship between the graphs of $f$, $f'$ and $f''$.

Use of both forms of notation, $\dfrac{\text{d}^2y}{\text{d}x^2}$ and $f''(x)$.

Technology can be used to explore graphs and calculate the derivatives of functions.

Link to: function graphing skills (SL 2.3).

Local maximum and minimum points.

Testing for maximum and minimum.

Using change of sign of the first derivative or using sign of the second derivative where $f''(x)>0$ implies a minimum and $f''(x)<0$ implies a maximum.

Optimization.

Examples of optimization may include profit, area and volume.

Points of inflexion with zero and non-zero gradients.

At a point of inflexion, $f''(x)=0$ and changes sign (concavity change), for example $f''(x)=0$ is not a sufficient condition for a point of inflexion for $y=x^4$ at $(0,0)$.

Use of the terms “concave-up” for $f''(x)>0$, and “concave-down” for $f''(x)<0$.

Kinematic problems involving displacement $s$, velocity $v$, acceleration $a$ and total distance travelled.

$v=\dfrac{\text{d}s}{\text{d}t}$; $a=\dfrac{\text{d}v}{\text{d}t}=\dfrac{\text{d}^2s}{\text{d}t^2}$

Displacement from $t_1$ to $t_2$ is given by $\displaystyle\int_{t_1}^{t_2}v(t) \text{d}t$.

Distance between $t_1$ to $t_2$ is given by $\displaystyle\int_{t_1}^{t_2}|v(t)| \text{d}t$.

Speed is the magnitude of velocity.

Indefinite integral of $x^n, (n∈ℚ)$, $\sin{x}$, $\cos{x}$, $\dfrac{1}{x}$ and $e^x$.

$\displaystyle\int\dfrac{1}{x}\text{d}x=\ln{x}+C$

The composites of any of these with the linear function $ax+b$.

Example:

$f'(x)=\cos{(2x+3)} ⇒ f(x)=\dfrac{1}{2}\sin{(2x+3)}+C$

Integration by inspection (reverse chain rule) or by substitution for expressions of the form:

$\displaystyle\int kg'(x) f(g(x)) \text{d}x$.

Examples:

• $\displaystyle\int2x(x^2+1)^4\text{d}x$,
• $\displaystyle\int4x\sin{x^2}\text{d}x$,
• $\displaystyle\int\dfrac{\sin{x}}{\cos{x}}\text{d}x$.

Definite integrals, including analytical approach.

$\displaystyle\int_a^b g'(x) \text{d}x=g(b)−g(a)$.

The value of some definite integrals can only be found using technology.

Link to: definite integrals using technology (SL 5.5).

Areas of a region enclosed by a curve $y=f(x)$ and the $x$-axis, where $f(x)$ can be positive or negative, without the use of technology.

Areas between curves.

Students are expected to first write a correct expression before calculating the area.

Technology may be used to enhance understanding of the relationship between integrals and areas.

Informal understanding of continuity and differentiability of a function at a point.

In examinations, students will not be asked to test for continuity and differentiability.

Understanding of limits (convergence and divergence).

Definition of derivative from first principles

$f'(x)=\lim\limits_{h→0} \dfrac{f(x+h)−f(x)}{h}$.

Link to: infinite geometric sequences (SL1.8).

Use of this definition for polynomials only.

Higher derivatives.

Familiarity with the notations $\dfrac{\text{d}^ny}{\text{d}x^n}$, $f^{(n)}(x)$.

Link to: proof by mathematical induction (AHL 1.15).

The evaluation of limits of the form $\lim\limits_{x→a} \dfrac{f(x)}{g(x)}$ and $\lim\limits_{x→∞} \dfrac{f(x)}{g(x)}$ using l’Hôpital’s rule or the Maclaurin series.

The indeterminate forms $\dfrac{0}{0}$ and $\dfrac{∞}{∞}$.

For example: $\lim\limits_{θ→0} \dfrac{\sin{θ}}{θ}=1$.

Link to: horizontal asymptotes (SL 2.8) .

Repeated use of l’Hôpital’s rule.

Implicit differentiation.

Related rates of change.

Optimisation problems.

Appropriate use of the chain rule or implicit differentiation, including cases where the optimum solution is at the end point.

Derivatives of $\tan{x}$, $\sec{x}$, $\cosec{x}$, $\cot{x}$, $a^x$, $\log_ax$, $\arcsin{x}$, $\arccos{x}$, $\arctan{x}$.

Indefinite integrals of the derivatives of any of the above functions.

The composites of any of these with a linear function.

Indefinite integral interpreted as a family of curves.

Examples:

• $\displaystyle\int\dfrac{1}{x^2+2x+5}\text{d}x=12\arctan{\dfrac{(x+1)}{2}}+C$
• $\displaystyle\int\sec^2{(2x+5)}\text{d}x=12\tan{(2x+5)}+C$

Use of partial fractions to rearrange the integrand.

$\displaystyle\int\dfrac{1}{x^2+3x+2}\text{d}x=\ln{\bigg|\dfrac{x+1}{x+2}\bigg|}+C$

Link to: partial fractions (AHL 1.11)

Integration by substitution.

On examination papers, substitutions will be provided if the integral is not of the form

$\displaystyle\int{kg'(x)f(g(x))\text{d}x}$.

Link to: integration by substitution (SL 5.10).

Integration by parts.

Examples:

• $\displaystyle\int{x\sin{x}\text{d}x}$,
• $\displaystyle\int{\ln{x}\text{d}x}$,
• $\displaystyle\int{\arcsin{x}\text{d}x}$

Repeated integration by parts.

Examples:

• $\displaystyle\int{x^2e^x\text{d}x}$ and
• $\displaystyle\int{e^x\sin{x}\text{d}x}$.

Area of the region enclosed by a curve and the $y$-axis in a given interval.

Volumes of revolution about the $x$-axis or $y$-axis.

First order differential equations.

Numerical solution of $\dfrac{\text{d}y}{\text{d}x}=f(x,y)$

using Euler’s method.

$x_{n+1}=x_n+h$, where $h$ is a constant.

Variables separable.

Example: the logistic equation

$\dfrac{\text{d}n}{\text{d}t}=kn(a−n), a, k∈ℝ$

Link to: partial fractions (AHL 1.11) and use of partial fractions to rearrange the integrand (AHL 5.15).

Homogeneous differential equation $\dfrac{\text{d}y}{\text{d}x}=f\Big(\dfrac{y}{x}\Big)$using the substitution $y=vx$.

Solution of $y'+P(x)y=Q(x)$, using the integrating factor.

Maclaurin series to obtain expansions for $e^x$, $\sin{x}$, $\cos{x}$, $\ln{(1+x)}$, $(1+x)^p, p∈ℚ$.

Use of simple substitution, products, integration and differentiation to obtain other series.

Example: for substitution: replace $x$ with $x^2$ to define the Maclaurin series for $e^{x^2}$.

Example: the expansion of $e^x\sin{x}$.

Maclaurin series developed from differential equations.