United Grade 12 AP Calculus AB

Chapters: 8

1. Limits and Continuity

Introducing Calculus: Can Change Occur at an Instant? · Defining Limits and Using Limit Notation · Estimating Limit Values from Graphs · Estimating Limit Values from Tables · Determining Limits Using Algebraic Properties of Limits · Determining Limits Using Algebraic Manipulation · Selecting Procedures for Determining Limits · Determining Limits Using the Squeeze Theorem · Connecting Multiple Representations of Limits · Exploring Types of Discontinuities · Defining Continuity at a Point · Confirming Continuity over an Interval · Removing Discontinuities · Connecting Infinite Limits and Vertical Asymptotes · Connecting Limits at Infinity and Horizontal Asymptotes · Working with the Intermediate Value Theorem (IVT)

Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.

2. Differentiation: Definition and Fundamental Properties

Defining Average and Instantaneous Rates of Change at a Point · Defining the Derivative of a Function and Using Derivative Notation · Estimating Derivatives of a Function at a Point · Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist · Applying the Power Rule · Derivative Rules: Constant, Sum, Difference, and Constant Multiple · Derivatives of cos x, sin x, ex, and ln x · The Product Rule · The Quotient Rule · Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Derivatives (Class 11): Slope, Rate of Change and Rules – A derivative tells how fast something changes at one moment. On a graph it is the slope of the tangent line. We find it with a limit: take a tiny step h, find the average change, and let h go to 0. Then we learn the quick rules for xⁿ, sin x, cos x, sums, differences, products and quotients.
Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.

3. Differentiation: Composite, Implicit, and Inverse Functions

The Chain Rule · Implicit Differentiation · Differentiating Inverse Functions · Differentiating Inverse Trigonometric Functions · Selecting Procedures for Calculating Derivatives · Calculating Higher Order Derivatives

Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.

4. Contextual Applications of Differentiation

Interpreting the Meaning of the Derivative in Context · Straight-Line Motion: Connecting Position, Velocity, and Acceleration · Rates of Change in Applied Contexts Other Than Motion · Introduction to Related Rates · Solving Related Rates Problems · Approximating Values of a Function Using Local Linearity and Linearization · Using L'Hospital's Rule for Determining Limits of Indeterminate Forms

Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
Linearization, Implicit Curves and Vector-Valued Derivatives – Zoom in on a smooth curve and it looks like a straight line: its tangent. The tangent line L(x) = f(a) + f′(a)(x − a) gives quick estimates near x = a. For curves like x² + y² = 25 we differentiate both sides and solve for dy/dx; a zero numerator gives a horizontal tangent, a zero denominator a vertical one. A moving point r(t) = (x(t), y(t)) has velocity r′(t) = (x′(t), y′(t)) and speed |r′(t)|. The nth term test says: if the terms of a series do not go to 0, the series diverges.
Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.

5. Analytical Applications of Differentiation

Using the Mean Value Theorem · Extreme Value Theorem, Global Versus Local Extrema, and Critical Points · Determining Intervals on Which a Function Is Increasing or Decreasing · Using the First Derivative Test to Determine Relative (Local) Extrema · Using the Candidates Test to Determine Absolute (Global) Extrema · Determining Concavity of Functions over Their Domains · Using the Second Derivative Test to Determine Extrema · Sketching Graphs of Functions and Their Derivatives · Connecting a Function, Its First Derivative, and Its Second Derivative · Introduction to Optimization Problems · Solving Optimization Problems · Exploring Behaviors of Implicit Relations

Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
Linearization, Implicit Curves and Vector-Valued Derivatives – Zoom in on a smooth curve and it looks like a straight line: its tangent. The tangent line L(x) = f(a) + f′(a)(x − a) gives quick estimates near x = a. For curves like x² + y² = 25 we differentiate both sides and solve for dy/dx; a zero numerator gives a horizontal tangent, a zero denominator a vertical one. A moving point r(t) = (x(t), y(t)) has velocity r′(t) = (x′(t), y′(t)) and speed |r′(t)|. The nth term test says: if the terms of a series do not go to 0, the series diverges.

6. Integration and Accumulation of Change

Exploring Accumulations of Change · Approximating Areas with Riemann Sums · Riemann Sums, Summation Notation, and Definite Integral Notation · The Fundamental Theorem of Calculus and Accumulation Functions · Interpreting the Behavior of Accumulation Functions Involving Area · Applying Properties of Definite Integrals · The Fundamental Theorem of Calculus and Definite Integrals · Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation · Integrating Using Substitution · Integrating Functions Using Long Division and Completing the Square · Selecting Techniques for Antidifferentiation

Integration as Accumulation: Riemann Sums to Volumes – A definite integral adds up a rate. Cut the region into thin rectangles (a Riemann sum), let them get thinner, and the sum becomes the exact area. The same idea gives total change, average value, displacement and distance, and the volume of solids made by spinning or stacking slices.
Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.

7. Differential Equations

Modeling Situations with Differential Equations · Verifying Solutions for Differential Equations · Sketching Slope Fields · Reasoning Using Slope Fields · Finding General Solutions Using Separation of Variables · Finding Particular Solutions Using Initial Conditions and Separation of Variables · Exponential Models with Differential Equations

Differential Equations – A differential equation connects a function y with its derivatives. Its order is the highest derivative present and its degree is the power of that derivative (when the equation is a polynomial in derivatives). A general solution has arbitrary constants; a condition like y(0) = 1 fixes them to give a particular solution. Class 12 solves first-order equations of three kinds: variables separable, homogeneous (put y = vx) and linear dy/dx + Py = Q (multiply by the integrating factor e^∫P dx).
Slope Fields, Euler's Method and Logistic Models – A differential equation dy/dx = f(x, y) gives the slope at every point. Drawing tiny lines with those slopes makes a slope field, and solution curves follow the lines. Euler's method walks along the field in straight steps: yₙ₊₁ = yₙ + h·f(xₙ, yₙ). The logistic model dP/dt = kP(1 − P/K) grows fastest at P = K/2 and levels off at the carrying capacity K. A falling object with drag, m dv/dt = mg − bv, approaches the terminal velocity v_t = mg/b.

8. Applications of Integration

Finding the Average Value of a Function on an Interval · Connecting Position, Velocity, and Acceleration of Functions Using Integrals · Using Accumulation Functions and Definite Integrals in Applied Contexts · Finding the Area Between Curves Expressed · Finding the Area Between Curves Expressed · Finding the Area Between Curves That Intersect at More Than Two Points · Volumes with Cross Sections: Squares and Rectangles · Volumes with Cross Sections: Triangles and Semicircles · Volume with Disc Method: Revolving · Volume with Disc Method: Revolving Around Other Axes · Volume with Washer Method: Revolving · Volume with Washer Method: Revolving Around Other Axes

Integration as Accumulation: Riemann Sums to Volumes – A definite integral adds up a rate. Cut the region into thin rectangles (a Riemann sum), let them get thinner, and the sum becomes the exact area. The same idea gives total change, average value, displacement and distance, and the volume of solids made by spinning or stacking slices.
Application of Integrals: Area Under Curves – The area between a curve y = f(x), the x-axis and the lines x = a and x = b is ∫ₐᵇ |f(x)| dx: we add thin vertical strips of height y and width dx. For curves given as x = g(y) we use horizontal strips. Symmetry saves work: find one part of a circle, parabola or ellipse and multiply. A circle of radius r gives πr² and an ellipse with semi-axes a, b gives πab.