BC Calculus Review: Differentiation, Applications, and Graphical Analysis

Review of Differentiation and Its Applications

Differentiation of Functions

Differentiation is a fundamental concept in calculus, used to determine the rate at which a function changes. The derivative of a function at a point gives the slope of the tangent line to the graph at that point.

• Definition: The derivative of a function $f(x)$ is defined as $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.

• Common Derivatives:

  • $\frac{d}{dx}(\sin x) = \cos x$

  • $\frac{d}{dx}(\tan x) = \sec^2 x$

  • $\frac{d}{dx}(e^x) = e^x$

  • $\frac{d}{dx}(\ln x) = \frac{1}{x}$

• Example: Differentiate $y = x^2 \sin(x)$:

  • Use the product rule: $\frac{d}{dx}(uv) = u'v + uv'$

  • $\frac{d}{dx}(x^2 \sin x) = 2x \sin x + x^2 \cos x$

Implicit Differentiation

Implicit differentiation is used when a function is not given explicitly as $y = f(x)$, but rather as a relation between $x$ and $y$.

• Definition: Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

• Example: For $x^2 + y^2 = 1$, differentiate both sides:

  • $2x + 2y \frac{dy}{dx} = 0$

  • $\frac{dy}{dx} = -\frac{x}{y}$

Applications of Derivatives

Derivatives are used to solve real-world problems involving rates of change, optimization, and motion.

• Related Rates: Problems where two or more quantities are related and change with respect to time.

  • Example: Water entering an inverted cone at a constant rate. Use geometry and related rates to find the rate at which the water level rises.

• Tangent and Normal Lines: The equation of the tangent line at $x=a$ is $y = f(a) + f'(a)(x-a)$.

• Optimization: Find maximum or minimum values by setting $f'(x) = 0$ and analyzing critical points.

Graphical Analysis and Interpretation

Critical Points and Inflection Points

Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined. Inflection points occur where $f''(x) = 0$ and the concavity changes.

• Absolute Maximum/Minimum: The highest/lowest value of $f(x)$ on a given interval.

• Relative Maximum/Minimum: Local highest/lowest points.

• Inflection Point: Where the graph changes concavity.

• Example: Given a table of $f(x)$, $f'(x)$, and $f''(x)$, identify where $f(x)$ attains maxima/minima and inflection points.

Graphing from Derivative Information

Given the graph of $f'(x)$, one can deduce properties of $f(x)$ such as increasing/decreasing intervals, maxima/minima, and concavity.

• Horizontal Tangent: Occurs where $f'(x) = 0$.

• Concavity: $f''(x) > 0$ implies concave up; $f''(x) < 0$ implies concave down.

• Example: Sketch the graph of $f(x)$ given the graph of $f'(x)$, noting where $f'(x)$ crosses the $x$-axis (critical points) and where $f'(x)$ changes sign (maxima/minima).

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as $x$ approaches a particular value.

• Definition: $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ gets close to $a$.

• Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$

• Continuity: A function is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

Table: Properties of a Function and Its Derivatives

The following table summarizes the behavior of a function $f(x)$ and its derivatives over different intervals:

x	0	0 < x < 1	1	1 < x < 2	2	2 < x < 3
f(x)	1	Positive	0	Negative	-1	Negative
f'(x)	Undefined	Positive	0	Negative	Undefined	Positive
f''(x)	Undefined	Positive	0	Negative	Undefined	Negative

Motion Along a Line: Position, Velocity, and Acceleration

Particle Motion

When a particle moves along a line, its position, velocity, and acceleration can be described using calculus.

• Position Function: $x(t)$ gives the location at time $t$.

• Velocity: $v(t) = x'(t)$

• Acceleration: $a(t) = x''(t)$

• Example: For $x(t) = 2t^3 - 6t^2$, find when the particle moves left ($v(t) < 0$), and its acceleration at $t=0$.

Special Topics and Problem Types

Symmetric Difference Quotient

The symmetric difference quotient is an alternative to the standard difference quotient for estimating derivatives:

• $\frac{f(x+h) - f(x-h)}{2h}$

• Used for numerical approximation and analysis of differentiability.

Discontinuities and Graph Sketching

Discontinuities occur where a function is not continuous. Sketching graphs with specified discontinuities and behavior is a common exercise.

• Jump Discontinuity: The function "jumps" from one value to another.

• Removable Discontinuity: A hole in the graph that can be "filled" by redefining the function.

• Example: Sketch a function with a cusp at $x=1$ and a discontinuity at $x=2$.

Summary of Key Concepts

• Differentiation is used to find rates of change and slopes of tangent lines.

• Implicit differentiation is necessary when functions are not given explicitly.

• Critical points and inflection points are found using first and second derivatives.

• Graphical analysis of $f'(x)$ provides insight into the behavior of $f(x)$.

• Limits and continuity are foundational for understanding differentiability.

• Applications include related rates, optimization, and motion problems.

Additional info: Some problems require calculator use for numerical approximation, and some involve advanced applications such as implicit differentiation for rotated ellipses and analysis of motion under gravity.