♾️AP Calculus AB/BC AP Cram Sessions 2021

Key Concepts and Definitions

• Limit describes the value a function approaches as the input approaches a certain value
• Continuity means a function has no breaks, gaps, or jumps in its graph
• Derivative measures the instantaneous rate of change of a function at a given point
  • First derivative represents the slope of the tangent line at a point
  • Second derivative indicates the concavity of the function
• Integral represents the area under a curve between two points
  • Definite integral has fixed upper and lower limits
  • Indefinite integral results in a function (antiderivative) without specific limits
• Riemann sum approximates the area under a curve using rectangles
• Fundamental Theorem of Calculus connects derivatives and integrals, showing they are inverse operations

Fundamental Theorems and Principles

• Intermediate Value Theorem states that if a function is continuous on a closed interval $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists a value $c$ in $[a, b]$ such that $f(c) = k$
• Extreme Value Theorem guarantees that a continuous function on a closed interval attains both a maximum and minimum value
• Rolle's Theorem specifies that if a function is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists a point $c$ in $(a, b)$ such that $f'(c) = 0$
• Mean Value Theorem extends Rolle's Theorem, stating that under the same conditions, there exists a point $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Fundamental Theorem of Calculus, Part 1 states that if $f$ is continuous on $[a, b]$, then the function $g(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$, with $g'(x) = f(x)$
• Fundamental Theorem of Calculus, Part 2 states that if $f$ is continuous on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$

Essential Formulas and Equations

• Power Rule for derivatives: $\frac{d}{dx} x^n = nx^{n-1}$
• Chain Rule for derivatives: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
• Product Rule for derivatives: $\frac{d}{dx} (f(x) \cdot g(x)) = f'(x)g(x) + f(x)g'(x)$
• Quotient Rule for derivatives: $\frac{d}{dx} (\frac{f(x)}{g(x)}) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$
• Integration by substitution: $\int f(g(x))g'(x) dx = \int f(u) du$, where $u = g(x)$
• Integration by parts: $\int u dv = uv - \int v du$
• Riemann sum formula: $\sum_{i=1}^n f(x_i^*) \Delta x$, where $x_i^*$ is a sample point in the $i$-th subinterval and $\Delta x$ is the width of each subinterval
• Arc length formula: $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$

Problem-Solving Strategies

• Identify the type of problem (derivative, integral, optimization, etc.) and choose an appropriate method
• Sketch a graph or diagram to visualize the problem, if applicable
  • For optimization problems, identify the function to be maximized or minimized and any constraints
• Break down complex problems into smaller, manageable steps
• Use substitution to simplify expressions or integrals
• Apply known formulas, theorems, and properties to solve the problem
• Check your answer for reasonableness and verify that it satisfies any given conditions
• If stuck, try a different approach or method
  • Consider using an alternate representation (e.g., graphical, numerical, or analytical)

Common Mistakes and Pitfalls

• Forgetting to apply the chain rule when differentiating composite functions
• Misapplying the quotient rule by incorrectly placing negative signs or not squaring the denominator
• Failing to adjust the limits of integration when making a substitution
• Incorrectly setting up or solving optimization problems by not identifying the correct function or constraints
• Misinterpreting the meaning of the second derivative in the context of concavity and inflection points
• Not checking for continuity or differentiability before applying certain theorems or properties
• Rounding too early in a calculation, leading to accumulated errors
• Neglecting to include the constant of integration ("+C") when finding indefinite integrals

Practice Problems and Solutions

1. Find $\frac{d}{dx} (x^3 + 2x^2 - 5x + 1)$
   • Solution: $3x^2 + 4x - 5$
2. Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
   • Solution: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x+2)(x-2)}{x-2} = \lim_{x \to 2} (x+2) = 4$
3. Find $\int (3x^2 + 2x - 1) dx$
   • Solution: $x^3 + x^2 - x + C$
4. Evaluate $\int_1^4 (2x + 1) dx$
   • Solution: $\int_1^4 (2x + 1) dx = [x^2 + x]_1^4 = (16 + 4) - (1 + 1) = 18$
5. A rectangular garden has a perimeter of 60 meters. Find the dimensions that maximize the garden's area.
   • Solution: Let $x$ be the width and $y$ be the length. Then $2x + 2y = 60$ and $y = 30 - x$. The area function is $A(x) = x(30-x) = 30x - x^2$. Setting $A'(x) = 0$ gives $30 - 2x = 0$, so $x = 15$. The dimensions that maximize the area are 15 meters by 15 meters.

Exam Tips and Techniques

• Read each question carefully and identify what is being asked
• Show all your work, as partial credit may be awarded for correct steps even if the final answer is incorrect
• Use proper notation and symbols throughout your solutions
• Manage your time wisely, allocating more time to challenging problems and less time to straightforward ones
• If you encounter a difficult problem, move on and return to it later if time allows
• Double-check your answers for accuracy and reasonableness before submitting your exam
• When graphing, label axes, key points, and asymptotes clearly
• If provided with a formula sheet, familiarize yourself with its contents before the exam to save time during the test

Advanced Topics and Extensions

• Parametric and polar equations and their derivatives
  • Parametric equations: $x = f(t)$, $y = g(t)$; $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$
  • Polar equations: $r = f(\theta)$; $\frac{dy}{dx} = \frac{dr/d\theta}{r}$
• Vector-valued functions and their derivatives and integrals
  • Position vector: $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$
  • Velocity vector: $\vec{v}(t) = \vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$
  • Acceleration vector: $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t) = \langle f''(t), g''(t), h''(t) \rangle$
• Partial derivatives and gradients for functions of several variables
  • Partial derivative with respect to $x$: $\frac{\partial f}{\partial x}$
  • Partial derivative with respect to $y$: $\frac{\partial f}{\partial y}$
  • Gradient: $\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle$
• Multiple integrals and their applications (volume, surface area, etc.)
  • Double integral: $\iint_R f(x, y) dA$
  • Triple integral: $\iiint_E f(x, y, z) dV$
• Differential equations and their solutions
  • Separable equations: $\frac{dy}{dx} = f(x)g(y)$; $\int \frac{1}{g(y)} dy = \int f(x) dx + C$
  • Linear first-order equations: $\frac{dy}{dx} + P(x)y = Q(x)$; solution: $y = e^{-\int P(x) dx} \left(\int Q(x)e^{\int P(x) dx} dx + C\right)$