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) = x^n, then f(x) = x^{n-1}

• Product Rule: If f(x) = g(x)*h(x), then f’(x) = g’(x)*h’(x) + g(x)*h(x)

• Quotient Rule: If \frac{g(x)}{h(x)} , then f’(x) = \frac{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:

• \frac{d}{dx}(e^x)= e^x

• \frac{d}{dx}(ln(x))=\frac{1}{x}

• \frac{d}{dx}(sin(x))=cos(x)

• \frac{d}{dx}(cos(x))=-sin(x)

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 \int 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 \int_{a}^{b}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)=\int_{a}^{x}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 \int_{b}^{a}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.