Calculus

Limits

A limit is the value that a function approaches as its input approaches a certain value. It is important to note that the limit is never actually reached, rather we just get infinitely close, and never go beyond it. For the scope of this course, you will mostly be working with an x-value as the input, and 0 or $±∞$ as the value being approached.


Notation:

$$\lim_{x \to a} f(x) = L$$

which is read as "the limit of $f(x)$ as $x$ approaches $a$", where $L$ is the value that $f(x)$ approaches as $x \to a$.


You may also see a $+$ or $-$ symbol superscript with the value being approached, which represents the side from which a value is being approached. For example, $\lim \limits_{x \to 0^+} f(x)$ is the limit of $f(x)$ as $x$ approaches zero from the positive (right) side.

Tangents, Normals, and Derivatives

Tangent: a straight line that touches a curve at one single point. At that point, the gradient of the curve is equal to the gradient of the tangent.

Normal: a straight line that is perpendicular to the tangent line.


Slope/Gradient of Curves:

In order to find the slope/gradient of a line, we need two points. So, if we have a point, $x$, we could choose another point, $x+h$ to calculate the gradient. However, as you may notice, for curved lines like a parabola there is no "gradient" of the entire curve - where instead we can find the "gradient" of one point on the curve, which will simply be the gradient of the tangent line at that point. This gradient will be very inaccurate if the second point on the curve is significantly larger than $x$, hence it is best for $x+h$ if $h$ is a very small value. In fact, if we just consider the limit of this gradient as $h$ approaches 0, we will find the correct gradient of the tangent line at the point $x$. This can be written as the following:

$$m=\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

where $m$ is the gradient of the line tangent to $f(x)$ at $x$. From this expression for the gradient of the tangent line, we get the notion of a derivative, where the derivative of a function is the gradient of the tangent line at a point on the function's curve. The value of the derivative of $f(x)$ at $x$ is simply the value you get for $m$ using the expression above.

Derivative Notation and Rules

Notation


There are two types of notation used for derivatives that IB expects you to know:


  • Lagrange Notation: this uses a "prime" or apostrophe symbol beside the function, where the derivative of $f(x)$ would be $f’(x)$. In order to write a specific value, just replace the letter for the input ($x$ in this case) with the value. Example: the derivative of $f(x)$ at $x=3$ becomes $f’(3)$.

  • Leibniz Notation: this notation expresses derivatives as $\frac{dy}{dx}$, indicating the derivative of $y$ with respect to $x$ where $y=f(x)$. In order to write a specific value for the input using this notation, you must write $\left.\frac{dy}{dx}\right\rvert_{x=a}$, where $a$ is your value. Example: the derivative of $f(x)$, where $y=f(x)$, at $x=3$ becomes $\left.\frac{dy}{dx}\right\rvert_{x=3}$.

Differentiation Rules


There are a few rules that you will need to know in order to differentiate functions. They were derived using the limit definition of the derivative above, and can be used to quickly calculate derivatives without using limits. These rules are:


  • Constant Rule: The derivative of a constant is always zero.
    Example: If $f(x) = 5$, then $f'(x) = 0$.

  • Power Rule: The derivative of $x$ raised to a power $n$ is $nx^{n-1}$.
    Example: If $f(x) = x^3$, then $f'(x) = 3x^2$.

  • Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their individual derivatives.
    Example: If $h(x) = f(x) + g(x)$, then $h'(x) = f'(x) + g'(x)$.

  • Constant Multiple Rule: A constant factor can be taken outside the derivative.
    Example: If $g(x) = 3f(x)$, then $g'(x) = 3f'(x)$.

  • Chain Rule: The derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.
    Example: If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.

  • Product Rule: The derivative of a product of two functions is the first function times the derivative of the second, plus the derivative of the first function times the second function.
    Example: If $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.

  • Quotient Rule: The derivative of a quotient of two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
    Example: If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$.

Other Important Derivative Rules:


  • Derivative of Sine: $\frac{d}{dx}(\sin x) = \cos x$

  • Derivative of Cosine: $\frac{d}{dx}(\cos x) = -\sin x$

  • Derivative of the Exponential Function: $\frac{d}{dx}(e^x) = e^x$

  • Derivative of a General Exponential Function: $\frac{d}{dx}(b^x) = b^x \ln(b)$, where $b > 0$

  • Derivative of the Natural Logarithm: $\frac{d}{dx}(\ln x) = \frac{1}{x}$

  • Derivative of a General Logarithm: $\frac{d}{dx}(\log_b x) = \frac{1}{x \ln(b)}$, where $b > 0$, $b \neq 1$

Higher Order Derivatives

If you take the derivative of a derivative function, you will get the second derivative, which is notated as:

$$f''(x) \qquad \text{or} \qquad \frac{d^2y}{dx^2}$$

This can be used to determine whether a point on a graph is a maximum, a minimum, or a point of inflection (see applications below). Similarly, you can take the derivative of a second derivative to get the third derivative, which is notated as:

$$f'''(x) \qquad \text{or} \qquad \frac{d^3y}{dx^3}$$

and so on, where the $n$th derivative is notated as:

$$f^{(n)}(x) \qquad \text{or} \qquad \frac{d^ny}{dx^n}$$

Applications of Derivatives

Turning Points:


There are two types of turning points:

  • 1. Local maxima
  • 2. Local minima

We know that when $f′(x) = 0$ there will be a local maximum or a minimum. Whether it is a maximum or minimum should be evident from looking at the graph of the original function. If a graph is not available, we can find out by plugging in a slightly smaller and slightly larger value than the point in question into $f′(x)$. If the smaller value is negative and the larger value positive, then it is a local minimum. If the smaller value is positive and the larger value negative, then it is a local maximum.


Furthermore, you can use the second derivative to tell whether a turning point is a local maximum or local minimum. When $f'(x)=0$, if $f''(x) < 0$ then we know that it is a local maximum (the derivative's value is decreasing). If $f''(x) > 0$ then we know that it is a local minimum (the derivative's value is increasing). If $f''(x) = 0$ then you cannot tell which one it is using the second derivative.



Points of Inflection:


An inflection (or inflexion) point is where there is a change in concavity, from up to down, or down to up. If the second derivative is changing sign, then that is the point where there is an inflection, and that means that the second derivative of the function is equal to zero or undefined.


There are some cases where $f''(x)=0$, but there is no change in concavity. However, this would not be an inflexion point, as inflexion points must meet both criteria.



Applications in Kinematics:


Kinematics is the branch of mechanics in physics concerned with the motion of objects. When you are given one function to calculate displacement, velocity, or acceleration you can use differentiation or integration (see further below) to determine the functions for the other two.


For example, velocity ($v$) is described as the rate of change of displacement ($s$) with respect to time. Hence, a velocity function is the derivative of the displacement function:

$$v(t) = \frac{ds}{dt} = s'(t)$$

Acceleration ($a$) is the rate of change of velocity, so the acceleration function is the derivative of the velocity function:

$$a(t) = \frac{dv}{dt} = v'(t)$$

This also means that acceleration is the second derivative of the displacement function:

$$a(t) = \frac{d^2s}{dt^2} = s''(t)$$



Optimization:


We can use differentiation to find minimum and maximum areas/volumes of various shapes. Often the key skill with these questions is to find an expression using simple geometric formulas and rearranging in order to differentiate.

Integration

Integration is essentially the opposite/inverse of differentiation. The notation for the integral of $f(x)$ is:

$$\int f(x) \, dx$$

where the $dx$ part simply means "with respect to $x$".



Constant of Integration:


If you consider that any constant disappears when differentiated, then when you are integrating, you must write $+ C$ at the end, in case there was any constant in the original function. If you want to find this mystery constant (i.e. find the actual antiderivative function), then you must know a set of coordinates that fit the original function. Then you can plug them in and solve for $C$.



Integration Rules


There are a few rules that you will need to know in order to integrate functions. They were derived working backwards from the derivative rules defined earlier. These rules are:


  • Integral of $dx$: $\displaystyle\int dx = x + C$

  • Integral of a Constant: $\displaystyle\int a \, dx = ax + C$, where $a$ is a constant

  • Integral of $x$: $\displaystyle\int x \, dx = \frac{x^2}{2} + C$

  • Power Rule: $\displaystyle\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$

  • Constant Multiple Rule: $\displaystyle\int a f(x) \, dx = a \int f(x) \, dx$, where $a$ is a constant

  • Sum/Difference Rule: $\displaystyle\int [f(x) ± g(x)] \, dx = \int f(x) \, dx ± \int g(x) \, dx$

  • Integral of Sine: $\displaystyle\int \sin x \, dx = -\cos x + C$

  • Integral of Cosine: $\displaystyle\int \cos x \, dx = \sin x + C$

  • Integral of the Exponential Function: $\displaystyle\int e^x \, dx = e^x + C$

  • Integral of $\frac 1x$: $\displaystyle\int \frac{1}{x} \, dx = \ln x + C$

  • Integral of $f(ax+b)$: $\displaystyle\int f(ax+b) \, dx = \frac{1}{a} F(ax+b) + C$, where $F(x)$ is the integral of $f(x)$

(Note: there are no equivalent integral rules for the product* rule and quotient rule in differentiation)


* The product rule has a semi-equivalent rule in integration called “integration by parts”, which is not covered in this course

U-Substitution Method

U-substitution is a change of variables used to make an integral easier to evaluate. You choose a part of the integrand to be $u$, usually the inside function of a composition, and then express the rest of the integral in terms of $u$ and $du$. This is a very helpful technique when the integrand contains a composite function and when part of the integrand is the derivative (or close to the derivative) of another part.


In order to do this, you first choose a part of the function to be $u$, then you differentiate $u$ to get $du$. You can then rewrite the integral in terms of $u$ and $du$, and integrate in terms of $u$.



Example:

Use u-substitution to evaluate: $\displaystyle\int (2x)(x^2+1)^3 \, dx$


First, let $u = x^2 + 1$.

Then, $\frac{du}{dx} = 2x \implies du = 2x\,dx$.



Substitute into the integral:

$$\int (x^2+1)^3 \cdot 2x \, dx = \int u^3 \, du$$

Integrate using the power rule:

$$\int u^3 \, du = \frac{u^4}{4} + C$$

Substitute back $u = x^2 + 1$:

$$\int (2x)(x^2+1)^3 \, dx = \frac{(x^2+1)^4}{4} + C$$

Definite and Indefinite Integrals

Indefinite Integral: When you evaluate $\displaystyle\int f(x)\,dx$, the function you get as a result is called the indefinite integral.


Definite Integral: An indefinite integral is instead written in the form $\displaystyle\int_a^b f(x)\,dx$. Here, you get a number as a result of evaluating the integral over the interval $a \leq x \leq b$. This gives the area between $f(x)$ and the x-axis from $x=a$ to $x=b$. This concept comes from the idea of Riemann sums, which involves splitting up this area into thin rectangles and evaluating the limit as the width approaches zero.



Indefinite integrals are evaluated in the following way, where the values for $a$ and $b$ are substituted as x-values into your indefinite integral:

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

where $$F(x) = \int f(x)\,dx \qquad \text{ or } \qquad F'(x) = f(x)$$


Be careful: the order you substitute $a$ and $b$ into the indefinite integral is relevant for your answer.

$$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$

Applications of Integrals

Area Between Curves:


By determining a definite integral for a function, you can find the area beneath the curve that is between the two x-values indicated as its limits. The integral over a region below the x-axis gives a negative value for its area. You must take that value as a positive value to determine the area between a curve and the x-axis. Sketching the graph will show what part of the function lies below the x-axis.


$$\text{Area of $f(x)$} = \int_a^b |f(x)|\,dx$$


Using definite integrals you can also find the areas enclosed between curves:

$$\text{Area Between $f(x)$ and $g(x)$} = \int_a^b [f(x) - g(x)]\,dx$$



Kinematics:


Similar to derivatives, the concept of integration is also highly applicable to kinematics. Displacement is an object's change in position. In contrast, distance refers to the path taken/travelled by an object, and it takes into account all changes in displacement even negative changes (going backwards). As stated previously, velocity is the rate of change of displacement, or the derivative of displacement. Hence, displacement is the integral of velocity:


$$\text{displacement from $t_1$ to $t_2$} = \int_{t_1}^{t_2} v(t)\,dt$$


This means that the sum of the area under a velocity curve is displacement, which we get from subtracting the area below the x-axis from the area above the x-axis.


However, to deal with the distance travelled by an object, we have to consider all of the area under a velocity curve - regardless if it is positive or not. To do this, we can use the absolute value, hence distance can be written as:


$$\text{distance from $t_1$ to $t_2$} = \int_{t_1}^{t_2} |v(t)|\,dt$$