AP Calculus BC Study Guide

Unit 1: Limits & Continuity

Key Concepts & Definitions

• Before Calculus Perspective: Traditionally, change was measured using average rates (e.g., average speed). Calculus introduces the concept of instantaneous rate of change, which is fundamentally defined by limits.

• Limits: The value that a function or sequence "approaches" as the input approaches some value. Limits are crucial for defining derivatives and integrals.

• Limit Laws: Properties that allow limits to be broken down across arithmetic operations:

  • Sum Rule: $\lim<em>{x\to c} (f(x) + g(x)) = \lim</em>{x\to c} f(x) + \lim_{x\to c} g(x)$

  • Difference Rule: $\lim<em>{x\to c} (f(x) - g(x)) = \lim</em>{x\to c} f(x) - \lim_{x\to c} g(x)$

  • Product Rule: $\lim<em>{x\to c} (f(x) \cdot g(x)) = \lim</em>{x\to c} f(x) \cdot \lim_{x\to c} g(x)$

  • Quotient Rule: $\lim<em>{x\to c} \frac{f(x)}{g(x)} = \frac{\lim</em>{x\to c} f(x)}{\lim<em>{x\to c} g(x)}$ (provided $\lim</em>{x\to c} g(x) \neq 0$).

  • These laws simplify the evaluation of complex limits.

• Graphical Limit Evaluation: Observe the y-values as $x$ approaches a point from both the left and the right. If they approach the same value, the limit exists and equals that value. If they differ, oscillate, or approach infinity, the limit does not exist. (A diagram illustrating one-sided limits and a non-existent limit would be helpful here).

• Numerical Limit Evaluation: Construct a table of values for $f(x)$ as $x$ approaches the point from both sides (e.g., $c-0.1, c-0.01, c-0.001$ and $c+0.1, c+0.01, c+0.001$). If the function values approach a common number, that is the limit.

• Squeeze Theorem (Sandwich Theorem): If $f(x)$ is bounded between two functions $g(x)$ and $h(x)$ (i.e., $g(x) \le f(x) \le h(x)$ for all $x$ near $c$), and if $\lim<em>{x\to c} g(x) = L$ and $\lim</em>{x\to c} h(x) = L$, then $\lim_{x\to c} f(x) = L$.

• Types of Discontinuities:

  • Removable Discontinuity: A "hole" in the graph. The limit exists, but the function is either undefined at that point or defined at a different value. It can be made continuous by redefining the function value.

  • Jump Discontinuity: The function "jumps" from one value to another. The left-hand limit and right-hand limit exist but are not equal.

  • Infinite Discontinuity: Occurs at a vertical asymptote where the function's values become unbounded (approach $+\infty$ or $-\infty$).

• Continuity at a Point: A function $f(x)$ is continuous at a point $c$ if all three conditions are met:

  1. $f(c)$ is defined.

  2. $\lim_{x\to c} f(x)$ exists.

  3. $\lim_{x\to c} f(x) = f(c)$.

• Continuity over an Interval: A function is continuous over an interval if it is continuous at every point within that interval.

• Limits at Infinity: Analyze the behavior of a function as $x \to \infty$ or $x \to -\infty$. These limits help identify horizontal asymptotes and the long-term trends of the function.

• Intermediate Value Theorem (IVT): If a function $f$ is continuous on a closed interval $[a, b]$, and $y$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the open interval $(a, b)$ such that $f(c) = y$. (Intuition: If a continuous function starts at one y-value and ends at another, it must pass through every y-value in between.)

Equations & Notation

• General limit notation: $\lim_{x\to c} f(x) = L$

• One-sided limits: $\lim<em>{x\to c^-} f(x)$ (left-hand limit) and $\lim</em>{x\to c^+} f(x)$ (right-hand limit).

• Definition of continuity at a point $c$: $\lim_{x\to c} f(x) = f(c)$.

• IVT formal statement: If $f$ is continuous on $[a, b]$ and f(a) < y < f(b) (or f(b) < y < f(a)), then there exists $c \in (a, b)$ such that $f(c) = y$.

Examples

• Example 1 (Graphical Limit): Consider a function $f(x)$ where as $x \to 2^-$, $f(x) \to 4$, and as $x \to 2^+$, $f(x) \to 1$. In this case, $\lim_{x\to 2} f(x)$ does not exist. (A diagram here would show a jump discontinuity at $x=2$).

• Example 2 (Squeeze Theorem): Evaluate $\lim<em>{x\to 0} x^2 \sin(\frac{1}{x})$. Since $-1 \le \sin(\frac{1}{x}) \le 1$, we have $-x^2 \le x^2 \sin(\frac{1}{x}) \le x^2$. As $x \to 0$, $-x^2 \to 0$ and $x^2 \to 0$. By the Squeeze Theorem, $\lim</em>{x\to 0} x^2 \sin(\frac{1}{x}) = 0$.

• Example 3 (Continuity Check): Determine if $f(x) = \begin{cases} x^2 &amp; x &lt; 1 \ 2x-1 &amp; x \ge 1 \end{cases}$ is continuous at $x=1$.

  1. $f(1) = 2(1)-1 = 1$ (defined).

  2. $\lim<em>{x\to 1^-} f(x) = \lim</em>{x\to 1^-} x^2 = 1^2 = 1$.

  3. $\lim<em>{x\to 1^+} f(x) = \lim</em>{x\to 1^+} (2x-1) = 2(1)-1 = 1$.

     Since the one-sided limits are equal, $\lim_{x\to 1} f(x) = 1$ (limit exists).

  4. Since $\lim_{x\to 1} f(x) = f(1)$, the function is continuous at $x=1$.

• Example 4 (IVT Application): Show that $f(x) = x^3 - 4x + 1$ has a root in $[0, 1]$.

  1. $f(x)$ is a polynomial, so it's continuous everywhere, including $[0, 1]$.

  2. $f(0) = 0^3 - 4(0) + 1 = 1$.

  3. $f(1) = 1^3 - 4(1) + 1 = 1 - 4 + 1 = -2$.

     Since f(1) < 0 < f(0), by the IVT, there exists a $c \in (0, 1)$ such that $f(c) = 0$. This means there is a root between 0 and 1.

Connections & Extensions

• Limits are the bedrock of calculus. They allow us to move from average rates of change over finite intervals to instantaneous rates of change at specific points, leading to the derivative.

• They connect to the idea of approaching values without necessarily reaching them, which is vital for understanding functions at points where they might be undefined or behave unusually.

• Continuity is a desirable property for many functions, ensuring that graphs can be drawn without lifting the pencil. It's essential for theorems like the IVT and EVT (Extreme Value Theorem).

Common Mistakes & Tips

• Confusing Limit Value with Function Value: The limit as $x \to c$ does not depend on $f(c)$. The function might not be defined at $c$, or $f(c)$ might be different from the limit.

• Ignoring One-Sided Limits: For a two-sided limit to exist, the left-hand and right-hand limits must be equal.

• Algebraic Errors: Errors in factoring, simplifying rational expressions, or rationalizing can lead to incorrect limit values.

• Misapplying IVT: Remember that the IVT only applies to functions that are continuous on the closed interval. Always check this condition first.

• Indeterminate Forms: Recognize forms like $0/0$ or $\infty/\infty$ as requiring further analysis (e.g., algebraic manipulation, L'Hôpital's Rule).

Unit Summary

• Limits: Define instantaneous change; evaluated graphically, numerically, or algebraically. Key limit laws apply to arithmetic operations.

• Squeeze Theorem: Used to find limits of functions bounded by two other functions.

• Continuity: No breaks, holes, or jumps in the graph. Defined by three conditions at a point: $f(c)$ exists, $\lim<em>{x\to c} f(x)$ exists, and $\lim</em>{x\to c} f(x) = f(c)$.

• Discontinuities: Removable (hole), Jump (different one-sided limits), Infinite (vertical asymptote).

• IVT (Intermediate Value Theorem): For a continuous function on $[a, b]$, it takes on every value between $f(a)$ and $f(b)$.

Unit 2: Derivatives and Differentiation Basics

Key Concepts & Definitions

• Derivative Definition (Limit Form): The derivative of a function $f(x)$ at a point $x$ is the instantaneous rate of change of the function at that point. It is defined as the limit of the difference quotient:

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

    This is also known as the slope of the tangent line to the curve at $x$.

• Alternative Form of the Derivative: The derivative at a specific point $c$ can also be defined as:

  • $f'(c) = \lim_{x\to c} \frac{f(x) - f(c)}{x - c}$

• Notations for the Derivative: Several notations are used, depending on context and preference:

  • Lagrange's notation: $f'(x)$ (read "f prime of x")

  • Leibniz's notation: $\frac{dy}{dx}$ (read "dy dx" or "the derivative of y with respect to x") or $\frac{d}{dx}f(x)$

  • Newton's notation: $\dot{y}$ (often used for derivatives with respect to time)

  • Euler's notation: $D_x y$

• Geometric Meaning: The derivative $f'(x)$ represents the slope of the tangent line to the graph of $f(x)$ at the point $(x, f(x))$. (A diagram showing a curve, a secant line, and a tangent line as the secant points converge would be illustrative).

• Differentiability Implies Continuity: If a function is differentiable at a point, it must be continuous at that point. However, the converse is not true: a function can be continuous but not differentiable (e.g., at sharp corners, cusps, or vertical tangent lines).

• Basic Derivative Rules:

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

  • Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$.

  • Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$.

  • Power Rule: The derivative of $x^n$ is $n \cdot x^{n-1}$ for any real number $n$: $\frac{d}{dx}[x^n] = n x^{n-1}$.

• Higher Order Derivatives: These are derivatives of derivatives. The second derivative is the derivative of the first derivative, the third derivative is the derivative of the second, and so on.

  • Notation: $f''(x)$, $\frac{d^2y}{dx^2}$ for the second derivative; $f'''(x)$, $\frac{d^3y}{dx^3}$ for the third derivative; $f^{(n)}(x)$, $\frac{d^ny}{dx^n}$ for the $n$-th derivative.

Equations & Notation

• Derivative as a limit definition (already stated above).

• Power Rule: $\frac{d}{dx} (x^n) = n x^{n-1}$.

• Derivative of a constant: $\frac{d}{dx} (c) = 0$.

• Sum/Difference, Constant Multiple rules (stated above).

Examples

• Example 1 (Using Limit Definition): Find the derivative of $f(x) = x^2$ using the limit definition.

  • $f'(x) = \lim<em>{h\to 0} \frac{(x+h)^2 - x^2}{h} = \lim</em>{h\to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$

  • $f'(x) = \lim<em>{h\to 0} \frac{2xh + h^2}{h} = \lim</em>{h\to 0} (2x + h) = 2x + 0 = 2x$.

• Example 2 (Applying Power Rule): Find the derivative of $f(x) = 5x^3 - \frac{1}{x^2} + 7$.

  • Rewrite: $f(x) = 5x^3 - x^{-2} + 7$

  • $f'(x) = 5(3x^{3-1}) - (-2x^{-2-1}) + 0$

  • $f'(x) = 15x^2 + 2x^{-3} = 15x^2 + \frac{2}{x^3}$.

• Example 3 (Higher Order Derivative): Find the second derivative of $f(x) = \sin x$.

  • $f'(x) = \cos x$

  • $f''(x) = \frac{d}{dx}(\cos x) = -\sin x$.

Connections & Extensions

• The derivative is the fundamental tool for understanding how functions change. It forms the basis for analyzing motion, optimization, and curve behavior.

• The concept of local linearity allows us to approximate a curve with a straight line (its tangent line) very close to the point of tangency, which is key for numerical methods and linear approximations.

Common Mistakes & Tips

• Forgetting Units: In application problems, the units of the derivative are the units of $y$ per unit of $x$ (e.g., miles per hour, dollars per item).

• Algebraic Errors: Careless algebra when simplifying the difference quotient can lead to incorrect derivatives.

• Power Rule Pitfalls: Remember that Constant Multiple Rule, Sum/Difference Rule are applied first. Forgetting to rewrite roots as fractional exponents or fractions as negative exponents before applying the power rule.

• Differentiability vs. Continuity: While differentiability implies continuity, be mindful that a continuous function might not be differentiable at points like sharp corners ($|x|$ at $x=0$) or vertical tangent lines ($x^{1/3}$ at $x=0$).

Unit Summary

• Definition: Derivative is the instantaneous rate of change ($f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$).

• Geometric interpretation: Slope of the tangent line.

• Key Rules: Constant Rule ($c' = 0$), Power Rule ($(x^n)' = nx^{n-1}$), Sum/Difference Rule, Constant Multiple Rule.

• Relationship: Differentiability implies continuity. Continuity does not imply differentiability.

• Higher Derivatives: Successive derivatives ($f'', f'''$).

Unit 3: Advanced Differentiation Rules

Key Concepts & Definitions

• Product Rule: Used to find the derivative of a product of two differentiable functions. If $h(x) = u(x)v(x)$, then its derivative is:

  • $h'(x) = u'(x)v(x) + u(x)v'(x)$, or $(uv)' = u'v + uv'$

• Quotient Rule: Used to find the derivative of a ratio of two differentiable functions. If $h(x) = \frac{u(x)}{v(x)}$, then its derivative is:

  • $h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$, or $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

• Chain Rule: The most powerful rule for differentiating composite functions. If $y = f(g(x))$, then:

  • $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$ (Derivative of the "outer" function evaluated at the "inner" function, multiplied by the derivative of the "inner" function).

• Derivatives of Common Trigonometric Functions:

  • $\frac{d}{dx} \sin x = \cos x$

  • $\frac{d}{dx} \cos x = -\sin x$

  • $\frac{d}{dx} \tan x = \sec^2 x$

  • $\frac{d}{dx} \cot x = -\csc^2 x$

  • $\frac{d}{dx} \sec x = \sec x \tan x$

  • $\frac{d}{dx} \csc x = -\csc x \cot x$

• Derivatives of Exponential and Logarithmic Functions:

  • $\frac{d}{dx} e^x = e^x$

  • $\frac{d}{dx} a^x = a^x \ln a$

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

  • $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$

• Implicit Differentiation: A technique used when $y$ is not explicitly defined as a function of $x$ (e.g., $x^2 + y^2 = 25$). We differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ (so we apply the chain rule, multiplying derivatives of $y$ terms by $\frac{dy}{dx}$), and then solve for $\frac{dy}{dx}$.

• Derivative of Inverse Functions: If $f$ is a differentiable function with an inverse $f^{-1}$, and $f'(g(x)) \ne 0$ where $g(x) = f^{-1}(x)$, then:

  • $[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$

  • Alternatively, if $x = g(y)$ is the inverse function of $y = f(x)$, then $\frac{dy}{dx} = \frac{1}{dx/dy}$ (if $\frac{dx}{dy} \ne 0$).

• Derivatives of Inverse Trigonometric Functions:

  • $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1-x^2}}$

  • $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1-x^2}}$

  • $\frac{d}{dx} \arctan x = \frac{1}{1+x^2}$

  • \frac{d}{dx} \arccot x = -\frac{1}{1+x^2}

  • $\frac{d}{dx} \text{arcsec } x = \frac{1}{|x|\sqrt{x^2-1}}$

  • $\frac{d}{dx} \text{arccsc } x = -\frac{1}{|x|\sqrt{x^2-1}}$

• Higher Order Derivatives: Continuation of finding successive derivatives. For example, $\frac{d^2y}{dx^2}$ is the second derivative, obtained by differentiating $\frac{dy}{dx}$ with respect to $x$.

Equations & Notation

• Product Rule: $(uv)' = u'v + uv'$

• Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

• Derivative of an inverse function: $[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$

• All derivatives of common functions and inverse trigonometric functions as listed above.

Examples

• Example 1 (Product Rule): Find the derivative of $h(x) = x^2 \sin x$.

  • Let $u = x^2$ and $v = \sin x$. Then $u' = 2x$ and $v' = \cos x$.

  • $h'(x) = u'v + uv' = (2x)(\sin x) + (x^2)(\cos x) = 2x \sin x + x^2 \cos x$.

• Example 2 (Quotient Rule): Find the derivative of $f(x) = \frac{e^x}{x}$.

  • Let $u = e^x$ and $v = x$. Then $u' = e^x$ and $v' = 1$.

  • $f'(x) = \frac{e^x \cdot x - e^x \cdot 1}{x^2} = \frac{e^x(x-1)}{x^2}$.

• Example 3 (Chain Rule): Find the derivative of $y = \sin(x^3)$.

  • Let $f(u) = \sin u$ and $u = x^3$. Then $f'(u) = \cos u$ and $u' = 3x^2$.

  • $\frac{dy}{dx} = \cos(x^3) \cdot 3x^2 = 3x^2 \cos(x^3)$.

• Example 4 (Implicit Differentiation): Find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$.

  • Differentiate both sides with respect to $x$:

    • $\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)$

    • $2x + 2y \frac{dy}{dx} = 0$

    • $2y \frac{dy}{dx} = -2x$

    • $\frac{dy}{dx} = -\frac{x}{y}$.

• Example 5 (Derivative of Inverse Function): If $f(x) = x^3 + x$, find $(f^{-1})'(2)$.

  1. First, find $f^{-1}(2)$. Let $y=2$. $2 = x^3 + x$. By inspection, $x=1$ is the solution, so $f^{-1}(2) = 1$.

  2. Find $f'(x) = 3x^2 + 1$.

  3. Evaluate $f'(f^{-1}(2)) = f'(1) = 3(1)^2 + 1 = 4$.

  4. Then $(f^{-1})'(2) = \frac{1}{f'(f^{-1}(2))} = \frac{1}{4}$.

Connections & Extensions

• These rules allow us to differentiate virtually any combination of standard functions, enabling the analysis of more complex real-world scenarios.

• Implicit differentiation is critical for working with relations that are not explicitly functions, such as circles, ellipses, and other curves.

• The chain rule is paramount; it extends our ability to handle composite functions, which are ubiquitous in science and engineering (e.g., rate of change of volume of a balloon as a function of time when its radius is also changing with time).

Common Mistakes & Tips

• Chain Rule Omissions: Forgetting to apply the chain rule, especially when differentiating a standard function with a more complex argument (e.g., differentiating $\sin(2x)$ as $\cos(2x)$ instead of $2\cos(2x){}$).

• Implicit Differentiation Errors: Forgetting to multiply by $\frac{dy}{dx}$ when differentiating a term involving $y$. Treating $y$ as a constant.

• Product vs. Quotient Rule: Mixing up the formulas, especially the sign in the numerator of the quotient rule ($-$).

• Simplification before Differentiation: Sometimes, algebraic manipulation (e.g., expanding or combining terms) can simplify the function before differentiating, making the process easier.

• Memorizing Inverse Trig Derivatives: Many inverse trig derivatives have a negative counterpart (e.g., $\arcsin x$ vs. $\arccos x$). Focus on the positive versions and remember that the "co" functions (-cos, -cot, -csc) lead to negative derivatives.

Unit Summary

• Product Rule: $(uv)' = u'v + uv'$

• Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

• Implicit Differentiation: Differentiate both sides with respect to $x$, apply chain rule to $y$ terms ($\dots \frac{dy}{dx}$), then solve for $\frac{dy}{dx}$.

• Inverse Functions: $[f^{-1}]'(x) = \frac{1}{f'(f^{-1}(x))}$.

• Derivatives of common functions: Trig, exponential, log, and inverse trig functions.

Unit 4: Applications of Differentiation

Key Concepts & Definitions

• Motion Concepts (Position, Velocity, Acceleration):

  • Position: $s(t)$ or $x(t)$ (function of time).

  • Velocity: $v(t) = s'(t)$ (first derivative of position). Represents the rate of change of position and direction of motion.

  • Acceleration: $a(t) = v'(t) = s''(t)$ (first derivative of velocity, second derivative of position). Represents the rate of change of velocity.

  • Speed: $|v(t)|$ (magnitude of velocity). Always non-negative.

  • Speeding Up/Slowing Down:

    • Speeding Up: Velocity and acceleration have the same sign (v(t)a(t) > 0).

    • Slowing Down: Velocity and acceleration have opposite signs (v(t)a(t) < 0).

  • Change in Direction: Occurs when velocity changes sign ($v(t)=0$ and changes sign, or $v(t)$ is undefined and changes sign).

• Related Rates Problems: Involve finding the rate at which one quantity is changing based on the known rates of other related quantities. All rates are with respect to time ($\frac{d}{dt}$).

  • General Approach (Four Steps):

    1. Identify Knowns and Unknowns: List all given rates and quantities, and the rate you need to find. This should include identifying expressions for $\frac{dx}{dt}, \frac{dy}{dt}$, etc.

    2. Draw a Picture: If applicable, sketch a diagram and label variables clearly.

    3. Find a Non-Derivative Relation: Establish an equation (or equations) that relates the variables before differentiation (e.g., Pythagorean theorem, volume formula, similar triangles). This relation should hold true at all times relevant to the problem.

    4. Differentiate Implicitly: Differentiate the relation(s) with respect to time ($t$). Remember to use the chain rule for any variable that is a function of time (which is usually all of them).

    5. Substitute and Solve: Plug in the known values for the variables and rates at the specific moment in question, then solve for the unknown rate.

• Linear Approximations (Linearization/Tangent Line Approximation): Using the tangent line at a point (a) to approximate the function's value near that point. This is based on the idea that a differentiable function looks like a straight line when zoomed in very closely.

  • The equation of the tangent line at $x=a$ is given by $L(x) = f(a) + f'(a)(x-a)$.

  • For $x$ close to $a$, $f(x) \approx L(x)$.

• Mean Value Theorem (MVT): If $f(x)$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one number $c$ in $(a, b)$ such that the instantaneous rate of change at $c$ equals the average rate of change over the interval:

  • $f'(c) = \frac{f(b) - f(a)}{b - a}$

• Extreme Value Theorem (EVT): If a function $f$ is continuous on a closed interval $[a, b]$, then $f$ is guaranteed to attain both an absolute maximum value and an absolute minimum value on that interval.

• Critical Points: Points $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ is undefined. All local (and thus global) extrema occur at critical points.

• Local Extrema: Highest or lowest points in a specific neighborhood on the graph (local maximum or local minimum).

• Global (Absolute) Extrema: The highest or lowest points over the entire domain or a specified interval.

• First Derivative Test: Used to locate local extrema by examining the sign changes of $f'(x)$。

  • If $f'(x)$ changes from $+$ to $-$ at $c$, then $f(c)$ is a local maximum.

  • If $f'(x)$ changes from $-$ to $+$ at $c$, then $f(c)$ is a local minimum.

  • If $f'(x)$ does not change sign, there is no local extremum.

• Concavity: Describes the direction of the curve's opening.

  • Concave Up: The graph of $f(x)$ lies above its tangent lines; $f'(x)$ is increasing; f''(x) > 0.

  • Concave Down: The graph of $f(x)$ lies below its tangent lines; $f'(x)$ is decreasing; f''(x) < 0.

• Point of Inflection: A point on the graph where the concavity changes (from up to down or vice versa). Occurs where $f''(x) = 0$ or $f''(x)$ is undefined, and $f''(x)$ changes sign.

• Second Derivative Test: Used to classify critical points as local maxima or minima.

  • If $f'(c) = 0$ and f''(c) < 0, then $f(c)$ is a local maximum.

  • If $f'(c) = 0$ and f''(c) > 0, then $f(c)$ is a local minimum.

  • If $f'(c) = 0$ and $f''(c) = 0$ (or is undefined), the test is inconclusive; use the First Derivative Test.

• Optimization Problems: Involve finding the maximum or minimum value of a quantity (objective function) subject to certain constraints. Often modeled using derivatives.

  • General Approach:

    1. Understand the Problem: Read carefully and identify what quantity needs to be optimized (maximized or minimized).

    2. Draw a Diagram: If applicable, label all quantities that are changing.

    3. Introduce Variables and Formulate Equations: Assign variables, determine the objective function (the quantity to optimize), and identify constraint equations (relations between variables).

    4. Reduce to One Variable: Use the constraint equation(s) to express the objective function in terms of a single independent variable.

    5. Find Critical Points: Differentiate the objective function (now in a single variable) and set the derivative to zero or find where it's undefined.

    6. Test for Max/Min: Use the First Derivative Test or Second Derivative Test to confirm if the critical point corresponds to a maximum or minimum.

    7. Check Endpoints: If the domain is a closed interval, evaluate the objective function at the endpoints.

    8. Answer the Question: State the optimized value and/or the dimensions that yield it, including units.

Equations & Notation

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• Linearization: $L(x) = f(a) + f'(a)(x-a)$

• Mean Value Theorem: $f'(c) = \frac{f(b) - f(a)}{b - a}$

• Conditions for concavity: f''(x) > 0 (concave up), f''(x) < 0 (concave down).

Examples

• Example 1 (Motion): A particle moves along the x-axis with position $s(t) = t^3 - 6t^2 + 9t + 1$. When is the particle speeding up and slowing down?

  • $v(t) = s'(t) = 3t^2 - 12t + 9 = 3(t-1)(t-3)$.

  • $a(t) = v'(t) = 6t - 12 = 6(t-2)$.

  • Critical points for velocity: $t=1, t=3$. Critical point for acceleration: $t=2$.

  • Analyze signs of $v(t)$ and $a(t)$ on intervals:\

    | Interval | $v(t)$ | $a(t)$ | $v(t)a(t)$ | Motion |

    | :-------- | :------- | :------- | :-------- | :-------- |

    | $(-\infty, 1)$ | $+$ | $-$ | $-$ | slowing down |

    | $(1, 2)$ | $-$ | $-$ | $+$ | speeding up |

    | $(2, 3)$ | $-$ | $+$ | $-$ | slowing down |

    | $(3, \infty)$ | $+$ | $+$ | $+$ | speeding up |

• Example 2 (Related Rates): A ladder 10 ft long is leaning against a wall. If the bottom of the ladder is sliding away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 6 ft from the wall?

  1. Known: $\frac{dx}{dt} = 1$ ft/s. Find: $\frac{dy}{dt}$ when $x=6$ ft.

  2. Diagram: Right triangle with legs $x$ (distance from wall to ladder base) and $y$ (height of ladder top on wall), hypotenuse 10.

  3. Relation: $x^2 + y^2 = 10^2 = 100$.

  4. Differentiate: $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$

  5. Substitute and Solve: When $x=6$, $6^2 + y^2 = 100 \implies y = 8$. Plug values in:

     • $2(6)(1) + 2(8) \frac{dy}{dt} = 0$

     • $12 + 16 \frac{dy}{dt} = 0$

     • $\frac{dy}{dt} = -\frac{12}{16} = -\frac{3}{4}$ ft/s. The top of the ladder is sliding down at $\frac{3}{4}$ ft/s.

• Example 3 (Linear Approximation): Use a linear approximation to estimate $\sqrt{4.1}$.

  • Let $f(x)=\sqrt{x}$ and $a=4$. Then $f(4)=\sqrt{4}=2$.

  • $f'(x) = \frac{1}{2\sqrt{x}}$, so $f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4}$.

  • $L(x) = f(a) + f'(a)(x-a) = 2 + \frac{1}{4}(x-4)$.

  • For $x=4.1$, $L(4.1) = 2 + \frac{1}{4}(4.1-4) = 2 + \frac{1}{4}(0.1) = 2 + 0.025 = 2.025$.

  • So, $\sqrt{4.1} \approx 2.025$.

• Example 4 (MVT): Find $c$ for $f(x) = x^2 - 2x$ on $[1, 5]$.

  1. $f(x)$ is a polynomial, so it's continuous on $[1, 5]$ and differentiable on $(1, 5)$.

  2. Average rate of change: $\frac{f(5) - f(1)}{5 - 1} = \frac{(5^2 - 2(5)) - (1^2 - 2(1))}{4} = \frac{(25-10) - (1-2)}{4} = \frac{15 - (-1)}{4} = \frac{16}{4} = 4$.

  3. $f'(x) = 2x - 2$.

  4. Set $f'(c)$ equal to the average rate of change: $2c - 2 = 4 \implies 2c = 6 \implies c = 3$.

  5. Since $3 \in (1, 5)$, the value $c=3$ satisfies the MVT.

• Example 5 (Optimization): A farmer wants to fence a rectangular pen with 100 feet of fencing, using an existing wall for one side. What are the dimensions of the pen that maximize the area?

  1. Maximize Area $A$

  2. Diagram: Rectangle with sides $x$ (perpendicular to wall) and $y$ (parallel to wall). Wall serves one $y$ side.

  3. Objective: $A = xy$. Constraint: $2x + y = 100$.

  4. Reduce to one variable: From constraint, $y = 100 - 2x$. Substitute into Area: $A(x) = x(100 - 2x) = 100x - 2x^2$.

  5. Find critical points: $A'(x) = 100 - 4x$. Set $A'(x) = 0 \implies 100 - 4x = 0 \implies 4x = 100 \implies x = 25$.

  6. Test: $A''(x) = -4$. Since A''(25) = -4 < 0, this is a local maximum.

  7. Check domain: x > 0. From $y=100-2x$, we need 100-2x > 0 \implies 2x < 100 \implies x < 50. So, the interval is $(0, 50)$. Endpoints are not included as they result in zero area. Thus, $x=25$ is the global maximum.

  8. Answer: If $x=25$ feet, then $y = 100 - 2(25) = 50$ feet. Max area is $(25)(50) = 1250$ sq ft.

Connections & Extensions

• These applications demonstrate the practical power of derivatives in understanding the real world
  —from predicting trajectories (motion) to designing efficient systems (optimization).

• The MVT is a bridge between average and instantaneous rates, justifying that somewhere in a continuous, differentiable interval, the instantaneous rate must equal the average rate.

• Curve sketching (combining information from first and second derivatives about increasing/decreasing, local extrema, and concavity) provides a comprehensive understanding of a function's graph.

Common Mistakes & Tips

• Motion: Confusing displacement with total distance (covered in integration unit). Not properly analyzing signs of $v(t)$ and $a(t)$ for speeding up/slowing down. Missing cases where $v(t)$ could be undefined.

• Related Rates: Substituting numerical values into the general relation before differentiating. Forgetting the chain rule (e.g., $d/dt(r^2) = 2r \frac{dr}{dt}$).

• MVT: Forgetting to check the conditions of continuity on $[a, b]$ and differentiability on $(a, b)$.

• Optimization: Not verifying whether a critical point is a maximum or minimum, or overlooking endpoints for global extrema. Not expressing the final answer in terms of the original problem.

• Concavity/Inflection Points: Confusing where $f''(x)=0$ with a point of inflection. $f''(x)$ must change sign.

Unit Summary

• Motion: $s(t) \xrightarrow{\text{derivative}} v(t) \xrightarrow{\text{derivative}} a(t)$. Speeding up ($v, a$ same sign), slowing down ($v, a$ opposite signs).

• Related Rates: Establish relation, differentiate implicitly w.r.t time, substitute. (Diagrams helpful).

• Linear Approximation: $f(x) \approx f(a) + f'(a)(x-a)$.

• MVT: $f'(c) = \frac{f(b) - f(a)}{b - a}$ (conditions: continuous on closed, differentiable on open).

• Extrema: Local at critical points ($f'(x)=0$ or undefined). Global on closed intervals (compare critical values and endpoints).

• Concavity: f''(x) > 0 (up), f''(x) < 0 (down). Inflection points where concavity changes.

• Optimization: Maximize/minimize objective using derivatives, subject to constraints.

Unit 5: Integrals and Basic Integration Techniques

Key Concepts & Definitions

• Antiderivative: A function $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$. If $F(x)$ is an antiderivative, then $F(x) + C$ (where $C$ is any constant) is the general antiderivative.

• Indefinite Integral: Represents the family of all antiderivatives of $f(x)$. It is denoted by $\int f(x) dx = F(x) + C$.

  • Properties of Indefinite Integrals:

    • Constant Multiple Rule: $\int k f(x) dx = k \int f(x) dx$

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

• Riemann Sums (Definition of Definite Integral): An approximation of the area under a curve by dividing an interval into subintervals and forming rectangles. The sum of the areas of these rectangles is a Riemann sum.

  • Types: Left Riemann Sum (LRAM), Right Riemann Sum (RRAM), Midpoint Riemann Sum (MRAM), Trapezoidal Sum (using trapezoids instead of rectangles). (Diagrams illustrating these would be helpful).

  • The definite integral is defined as the limit of these Riemann sums as the number of subintervals approaches infinity:

    • $\int<em>{a}^{b} f(x) dx = \lim</em>{n\to\infty} \sum<em>{i=1}^{n} f(x</em>i^<em>) \Delta x$ where $x_i^</em>$ is a point in the $i$-th subinterval and $\Delta x = \frac{b-a}{n}$.

• Definite Integral: Represents the net accumulated area between the function's graph and the x-axis over a given interval $[a, b]$. Areas above the x-axis are positive; areas below are negative.

  • Properties of Definite Integrals:

    • Order of Integration: $\int<em>{a}^{b} f(x) dx = -\int</em>{b}^{a} f(x) dx$

    • Zero Interval: $\int_{a}^{a} f(x) dx = 0$

    • Additivity: $\int<em>{a}^{c} f(x) dx + \int</em>{c}^{b} f(x) dx = \int_{a}^{b} f(x) dx$

• Fundamental Theorem of Calculus (FTC):

  • FTC Part 1: Relates differentiation and integration. If $f$ is continuous on $[a, b]$, then the function $g(x) = \int_{a}^{x} f(t) dt$ is differentiable on $(a, b)$ and $g'(x) = f(x)$. This means differentiation effectively "undoes" integration.

  • FTC Part 2: Provides a practical way to evaluate definite integrals. If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$), then:

    • $\int_{a}^{b} f(x) dx = F(b) - F(a)$

      This theorem transforms the problem of finding areas into the problem of finding antiderivatives.

• Technique: Substitution (u-substitution): A method for integrating composite functions, essentially the reverse of the chain rule. It involves making a substitution $u=g(x)$ so that $du = g'(x) dx$, transforming the integral into a simpler form in terms of $u$. After integrating with respect to $u$, substitute back $x$ to express the result in terms of the original variable.

  • For definite integrals, either change the limits of integration to be in terms of $u$ or substitute back to $x$ before evaluating.