Calculus Study Guide

Unit 1: Limits and Continuity

(Core Concepts & Formulas)

• Definition of a Limit: A limit is the value that a function approaches as the input variable gets closer and closer to a specific value.

  • Mathematically written as: limx→c​f(x)=L

  • Crucial Strategy: We do not care about the actual value of the function at x=c, only what value it approaches from the surrounding neighborhoods.

• Direct Substitution: For any continuous function (like simple polynomials), the limit can be found simply by evaluating the function at that point: limx→c​f(x)=f(c).

• Graphical & Tabular Estimation:

  • Graphs: Trace the curve from both the left and right sides toward the target x-value. If the left-hand limit (limx→c−​f(x)) does not equal the right-hand limit (limx→c+​f(x)), the limit Does Not Exist (DNE).

  • Tables: Look at the outputs for inputs extremely close to the target value (e.g., 2.999 and 3.001 when approaching 3).

• Algebraic Manipulation:

  • Used when direct substitution yields an indeterminate form like 00​.

  • Factoring: Factor out the numerator and denominator to cancel out removable discontinuities (holes).

  • Example: For (x+3)(x−3)(x+3)(x+2)​, the term (x+3) can be canceled out, revealing a hole at x=−3.

• The Squeeze (Sandwich) Theorem:

  • Conditions: If g(x)≤f(x)≤h(x) for all x in an open interval containing a (except possibly at a itself), and limx→a​g(x)=limx→a​h(x)=L.

  • Conclusion: Then limx→a​f(x)=L.

• Special Trigonometric Limits (Must Memorize):

  • limx→0​xsin(x)​=1

  • limx→0​xcos(x)−1​=0

  • limx→0​xsin(ax)​=a

  • limx→0​sin(bx)sin(ax)​=ba​

• Types of Discontinuities:

  • Removable Discontinuity: The graph has a single hole. It can be "removed" by redefining the function at that single point.

  • Jump Discontinuity: The curve abruptly breaks and starts at a different height. Left and right limits exist but are unequal.

  • Essential / Infinite Discontinuity: The curve approaches positive or negative infinity, creating a vertical asymptote.

• Three Conditions for Continuity: A function f(x) is continuous at a point x=c if and only if:

  1. f(c) is defined (the point exists).

  2. limx→c​f(x) exists (left-hand limit = right-hand limit).

  3. limx→c​f(x)=f(c) (the limit equals the point value).

  • Note: A function is continuous on an interval if it satisfies these conditions at every single point in the interval.

• Limits and Asymptotes:

  • Vertical Asymptote: Occurs if limx→c±​f(x)=±∞.

  • Horizontal Asymptote (End Behavior): Found by evaluating limx→∞​f(x) or limx→−∞​f(x).

• Horizontal Asymptote Rules for Rational Functions:

  • Top Heavy: If the highest degree is in the numerator, the limit is ±∞ (No horizontal asymptote).

  • Bottom Heavy: If the highest degree is in the denominator, the limit is 0 (y=0 is the asymptote).

  • Equal Degree: If degrees match, the limit is the ratio of the leading coefficients.

• Intermediate Value Theorem (IVT):

  • Condition: f(x) must be continuous on the closed interval [a,b].

  • Conclusion: If C is any number between f(a) and f(b), there must exist at least one number c in [a,b] such that f(c)=C.

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Unit 2: Differentiation: Definition and Fundamental Properties

(Core Concepts & Formulas)

• Average Rate of Change (AROC): The slope of a secant line connecting two points over an interval [x1​,x2​].

  AROC=x2​−x1​f(x2​)−f(x1​)​

• Instantaneous Rate of Change (IROC): The slope of the curve at a precise point in time, found by squeezing the interval of a secant line down to zero.

• Definition of the Derivative: The mathematical definition of IROC using a limit.

  f′(x)=h→0lim​hf(x+h)−f(x)​

  • Alternate Form (at a point c): f′(c)=limx→c​x−cf(x)−f(c)​

• Geometric Meaning: The derivative represents the slope of the tangent line to a curve at a single point.

• Notation Table:

  Function	First Derivative	Second Derivative
  f(x)	f′(x)	f′′(x)
  g(x)	g′(x)	g′′(x)
  y	y′ or dxdy​	y′′ or dx2d2y​

• Basic Derivative Rules:

  • Constant Rule: dxd​[k]=0

  • Constant Multiple Rule: dxd​[k⋅f(x)]=k⋅f′(x)

  • Power Rule: dxd​[xn]=n⋅xn−1 (Strategy: "Multiply down, decrease the power by one")

• Advanced Derivative Rules:

  • Product Rule: dxd​[u⋅v]=u⋅dxdv​+v⋅dxdu​ (Strategy: 1d2+2d1)

  • Quotient Rule: dxd​[vu​]=v2v⋅dxdu​−u⋅dxdv​​ (Strategy: "Low d-High minus High d-Low, over Low-squared")

• Trigonometric & Transcendental Memory Derivatives:

  • dxd​[sin(x)]=cos(x)

  • dxd​[cos(x)]=−sin(x)

  • dxd​[ex]=ex

  • dxd​[ln(x)]=x1​

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Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

核心概念与公式 (Core Concepts & Formulas)

• The Chain Rule: Used when differentiating composite functions (a function inside another function).

  dxd​[f(g(x))]=f′(g(x))⋅g′(x)

  • Strategy: "Derivative of the outside, leave the inside completely alone, then multiply by the derivative of the inside."

• Implicit Differentiation: Used when x and y are mixed together and y cannot easily be isolated.

  • Strategy: Differentiate both sides with respect to x. Every time you take the derivative of a term containing y, you must attach a dxdy​ via the chain rule. Then factor out and isolate dxdy​.

• Inverse Function Differentiation: If g(x) is the inverse of f(x), then their coordinates are flipped: if f(a)=b, then g(b)=a.

  g′(b)=f′(a)1​org′(x)=f′(g(x))1​

  • Strategy: To find the slope of the inverse function at a point, find the slope of the original function at its flipped buddy point and take its reciprocal.

• Inverse Trigonometric Derivatives:

  • dxd​[arcsin(x)]=1−x2​1​

  • dxd​[arccos(x)]=−1−x2​1​

  • dxd​[arctan(x)]=1+x21​

• Pro Tips for Derivative Success:

  • If you see a function nested within another, automatically use the Chain Rule.

  • If evaluating a derivative at a specific numerical point, plug the number in immediately after differentiating to avoid complex algebraic clean-up.

  • If asked to find a second derivative implicitly, simplify your first derivative expression as much as possible before starting the next round.

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Unit 4: Contextual Applications of Differentiation

核心概念与公式 (Core Concepts & Formulas)

• Interpreting derivatives: A derivative dxdy​ tells you how fast the output units (y) change per every single unit of input (x).

• Straight Line Motion (PVA):

  • Position: x(t) or s(t) → measures where the object is (meters).

  • Velocity: v(t)=x′(t) → measures speed and direction (m/s).

  • Acceleration: a(t)=v′(t)=x′′(t) → measures how fast velocity changes (m/s2).

• Speed Behavior Rule:

  • An object is speeding up if v(t) and a(t) share the same sign (both positive or both negative).

  • An object is slowing down if v(t) and a(t) have opposite signs.

• Related Rates: Problems where multiple variables change with respect to time (t).

  • Strategy Steps:

    1. Read carefully and identify given rates (e.g., dtdV​=10).

    2. Set up a geometric formula relating variables (e.g., Sphere Volume: V=34​πr3).

    3. Differentiate implicitly with respect to time t (e.g., dtdV​=4πr2dtdr​).

    4. Substitute known constants and solve for the missing rate. Include proper units!

• Linearization (Tangent Line Approximation): Using a tangent line at a known, easy value (x=a) to approximate a difficult nearby value (x+Δx).

  f(x+Δx)≈f(a)+f′(a)(x−a)

• L'Hospital's Rule: If evaluating a limit gives 00​ or ∞∞​, you can take the derivative of the top and bottom independently.

  If x→clim​g(x)f(x)​=00​, then x→clim​g(x)f(x)​=x→clim​g′(x)f′(x)​

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Unit 5: Analytical Applications of Differentiation

核心概念与公式 (Core Concepts & Formulas)

• Mean Value Theorem (MVT):

  • Conditions: f(x) must be continuous on [a,b] and differentiable on (a,b).

  • Conclusion: There must exist at least one point c in (a,b) where the instantaneous slope equals the average slope:

    f′(c)=b−af(b)−f(a)​

• Rolle's Theorem: A specific version of the MVT. If f(a)=f(b), there must be a point where f′(c)=0 (a flat spot).

• Extreme Value Theorem (EVT): If a function is continuous on a closed interval, it must have an absolute maximum and an absolute minimum value on that interval.

• Critical Points: Anywhere f′(x)=0 or where f′(x) is undefined. These are candidates for local extrema.

• First Derivative Test (Increasing/Decreasing):

  • If f′(x)>0, then f(x) is increasing.

  • If f′(x)<0, then f(x) is decreasing.

  • Relative Maximum: Occurs where f′(x) changes from positive to negative (+ to −).

  • Relative Minimum: Occurs where f′(x) changes from negative to positive (− to +).

• Absolute Extrema Candidate's Test: To find absolute maximums/minimums on a closed interval [a,b]:

  1. Find all critical numbers inside the interval.

  2. Create a T-chart listing endpoints a and b, alongside critical points.

  3. Evaluate these inputs in the ORIGINAL function f(x). The highest output is the absolute max; the lowest is the absolute min.

• Concavity and the Second Derivative:

  • If f′′(x)>0, the function is Concave Up (looks like a smile ⌣).

  • If f′′(x)<0, the function is Concave Down (looks like a frown ⌢).

  • Point of Inflection (POI): A point where f′′(x) changes signs (+ to − or − to +).

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Unit 6: Integration and Accumulation of Change

核心概念与公式 (Core Concepts & Formulas)

• The Definite Integral: Represents the total accumulated accumulation or the geometric area under a curvebounded by the x-axis.

• Riemann Sums: Approximating the area under a curve using shapes.

  • Left-Hand Sum: Uses the height of the left edge of each rectangle.

  • Right-Hand Sum: Uses the height of the right edge of each rectangle.

  • Midpoint Sum: Uses the value directly in the middle of each interval for the height.

  • Trapezoidal Sum: Uses trapezoids to fit curves smoothly. Formula for one subinterval: Area=21​(b1​+b2​)h, where h=Δx.

• Fundamental Theorem of Calculus (FTC Part 1):

  ∫ab​f(x)dx=F(b)−F(a)

  • Where F(x) is the antiderivative of f(x).

• The Indefinite Integral & Constant of Integration:

  ∫xndx=n+1xn+1​+C

  • Crucial Strategy: Always write +C for indefinite integrals because taking the derivative of any constant constant yields zero.

• Integration by U-Substitution (Reverse Chain Rule):

  • Strategy Steps:

    1. Identify an inner expression to call u whose derivative (du) is also present in the integral.

    2. Differentiate u to find dxdu​, and substitute to replace dx.

    3. Integrate with respect to u.

    4. Substitute the original expression back in for u.

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Unit 7: Differential Equations

核心概念与公式 (Core Concepts & Formulas)

• Differential Equations: An equation containing variables and their derivatives (e.g., dxdy​=2x). It describes how a system changes over time.

• Slope Fields: A visual map made of tiny slope lines indicating the steepness/direction of solutions at any coordinate (x,y).

  • Strategy: To draw one, plug (x,y) coordinates into the differential equation and draw a matching tick mark.

  • Solution Curves: Follow the flow lines smoothly like a boat drifting down a river current.

• Separable Differential Equations:

  • Strategy Method (SIPPY):

    • S - Separate: Get all y variables and dy on one side, and all x variables and dx on the other side.

    • I - Integrate: Take the integral of both sides.

    • P - Plus C: Add your +C immediately to the x-side.

    • P - Plug in: Use your initial condition point (e.g., y(0)=5) to solve for the numerical value of C.

    • Y - Y equals: Isolate y algebraically to state your final explicit solution.

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Unit 8: Applications of Integration

核心概念与公式 (Core Concepts & Formulas)

• Average Value of a Function: Gives the average height of a continuously changing function over an interval [a,b].

  Average Value=b−a1​∫ab​f(x)dx

• Net Change / PVA Integration Table:

  Goal	Integral Setup
  Displacement (Net change in position)	∫ab​v(t)dt
  Total Distance Traveled	$\int_{a}^{b}
  Final Position	x(b)=x(a)+∫ab​v(t)dt

• Area Between Two Curves:

  Area=∫ab​(Top Function−Bottom Function)dx

  • Strategy: Find intersection points a and b to establish your integration boundaries.

• Volume by Cross-Sections: Building 3D solids by piling up shapes over a base area bounded by functions.

  V=∫ab​A(x)dx

  • Common Area Formulas A(x):

    • Squares: Side2

    • Semicircles: 8π​(Diameter)2

• Volume of Revolution (Disc Method): Rotating a region around a flat axis to build a round, solid object.

  V=π∫ab​[R(x)]2dx

  • Where R(x) is the radius function extending from the axis of rotation to the boundary curve.

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