Calculus (IB)
Differentiation
1.1. Definition of the Derivative:
The derivative of a function represents the rate of change of that function at a given
point.
It measures how the function's output changes concerning its input.
1.2. Rules of Differentiation:
Power Rule: If f (x) = xn , then f (x) = xn−1
Product Rule: If f (x) = g(x) ∗ h(x), then f ’(x) = g’(x) ∗ h’(x) + g(x) ∗ h(x)
Quotient Rule: If h(x)
g(x)
, then f ’(x) = g’(x)∗h(x)−g(x)∗h’(x)
[h(x)] 2
Chain Rule: If f (x) = g(h(x)), then f (x) = g′ (h(x)) ∗ h′ (x)
′
1.3. Finding Derivatives of Elementary Functions:
Derivatives of basic functions like polynomials, trigonometric functions, exponential
functions, and logarithmic functions.
1.4. Applications of Derivatives:
Finding Maxima/Minima: Use derivatives to find critical points and determine whether
they correspond to maxima, minima, or points of inflection.
Optimization: Use derivatives to optimize functions, such as maximizing profit or
minimizing cost.
Related Rates: Use derivatives to solve problems involving rates of change in related
variables.
1.5. Notable Derivatives:
d x
(e ) = ex
dx
d 1
(ln(x)) =
dx x
d
(sin(x)) = cos(x)
dx
d
(cos(x)) = −sin(x)
dx
1.6. Implicit Differentiation:
Technique used to find the derivative of an implicitly defined function.
1.7. Higher Order Derivatives:
Second and higher derivatives denote rates of change of rates of change
(acceleration, jerk, etc.).
1.8. Derivatives in Graphical Analysis:
Interpretation of the slope of the tangent line as the derivative at a point.
Derivative as a measure of the rate of change of a function's graph.
Integration:
2.1. Indefinite Integrals:
The indefinite integral of a function represents a family of antiderivatives or primitives.
Denoted as ∫ f (x)dx, where f (x) is the integrand and d(x) indicates the variable of
integration.
2.2. Basic Definite Integrals and Their Properties:
The definite integral of a function over an interval represents the signed area between
the graph of the function and the x-axis.
Denoted as ∫ab f (x)dx, where a and b are the limits of integration.
Properties include linearity, the additive property, and the constant multiple property.
2.3. Integration Techniques:
Substitution: Involves substituting a new variable to simplify the integrand.
Integration by Parts: Involves applying the product rule for differentiation in reverse.
Partial Fractions: Used to integrate rational functions by decomposing them into
simpler fractions.
2.4. Applications of Integration:
Area under Curves: Use definite integrals to find the area between the curve and the x-
axis over a given interval.
Finding Volumes: Use definite integrals to find volumes of solids of revolution using
methods like disk/washer and shell methods.
2.5. Fundamental Theorem of Calculus (FTC):
Part 1: If f is continuous on [a, b] then the function F defined by F (x) = ∫ax f (t)dt is
continuous on (a, b) and differentiable on (a, b), and F ′ (x) = f (x)
Part 2: If F is an antiderivative of f on [a, b] then ∫ba f (x)dx = F (b) − F (a)
2.6. Applications of Integration in Geometry:
Arc Length: Use integrals to find the length of a curve.
Surface Area: Use integrals to find the surface area of a solid of revolution.
2.7. Numerical Integration:
Techniques such as the Trapezoidal Rule for approximating definite integrals when an
analytic solution is not feasible.
2.8. Integration in Graphical Analysis:
Interpretation of the integral as the accumulation of quantities represented by the
function's graph.
Applications of Differentiation and Integration:
3.1. Rate of Change Problems:
Definition: Use derivatives to analyze how one quantity changes concerning another.
Example: Speed of a moving object at a specific time, rate of change of population
growth, etc.
3.2. Optimization Problems:
Definition: Use derivatives to find maximum or minimum values of a function.
Example: Maximizing profit, minimizing cost, optimizing dimensions of a container, etc.
3.3. Area and Volume Problems:
Definition: Use integrals to find the area under a curve or the volume of a solid.
Example: Finding the area enclosed by a curve and the x-axis, calculating the volume
of a three-dimensional shape like a cylinder or cone.
3.4. Related Rates Problems:
Use derivatives to find the rate of change of one quantity with respect to another
related quantity.
Example: Rate at which the area of a circle is changing concerning its radius, rate of
change of the volume of a cone concerning its height, etc.
3.5. Motion Problems:
Definition: Use derivatives to analyze the motion of objects.
Example: Position, velocity, acceleration of an object at a given time, distance traveled
over a certain time interval, etc.
