التفاضل والتكامل AP (BC) | AP®︎ Calculus BC — مقرر…
مِسبار · مقرر جامعي مجاني
AP®︎ Calculus BC
تعلم التفاضل والتكامل AP (BC) (AP®︎ Calculus BC) مجانًا عبر 11 وحدات و456 فيديو تعليميًا، بإجمالي نحو 47 ساعة، ضمن مكتبة مِسبار الجامعية.
Learn AP®︎ Calculus BC—everything from AP®︎ Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP®︎ test.
- 11 وحدات.
- 456 فيديو.
- نحو 47 ساعة مشاهدة.
- المستوى: advanced-placement.
ابدأ مشاهدة المقرر داخل مكتبة مِسبار
التخصصات المرتبطة بالمقرر
- الهندسة الكهربائية (Electrical Engineering)
- هندسة الحاسوب (Computer Engineering)
- الهندسة الميكانيكية (Mechanical Engineering)
- الهندسة المدنية (Civil Engineering)
- الهندسة الكيميائية (Chemical Engineering)
- الهندسة الصناعية (Industrial Engineering)
- هندسة الطيران والفضاء (Aerospace Engineering)
- الذكاء الاصطناعي (Artificial Intelligence)
- علم البيانات (Data Science)
- الروبوتات (Robotics)
- الرياضيات (Mathematics)
- الفيزياء (Physics)
- العمارة (Architecture)
- تعليم الرياضيات (Mathematics Education)
وحدات ودروس المقرر
1. Limits and continuity
59 فيديو تعليمي.
موضوعات ودروس الوحدة
- Khan Academy in the classroom
- Sal interviews the AP Calculus Lead at College Board
- Limits intro
- Estimating limit values from graphs
- Unbounded limits
- One-sided limits from graphs
- One-sided limits from graphs: asymptote
- Connecting limits and graphical behavior
- Approximating limits using tables
- Estimating limits from tables
- One-sided limits from tables
- Limit properties
- Limits of combined functions
- Limits of combined functions: piecewise functions
- Theorem for limits of composite functions
- Theorem for limits of composite functions: when conditions aren't met
- Limits of composite functions: internal limit doesn't exist
- Limits of composite functions: external limit doesn't exist
- Limits by direct substitution
- Undefined limits by direct substitution
- Limits of trigonometric functions
- Limits of piecewise functions
- Limits of piecewise functions: absolute value
- Limits by factoring
- Limits by rationalizing
- Trig limit using Pythagorean identity
- Trig limit using double angle identity
- Strategy in finding limits
- Squeeze theorem intro
- Limit of sin(x)/x as x approaches 0
- Limit of (1-cos(x))/x as x approaches 0
- Types of discontinuities
- Continuity at a point
- Worked example: Continuity at a point (graphical)
- Worked example: point where a function is continuous
- Worked example: point where a function isn't continuous
- Continuity over an interval
- Functions continuous on all real numbers
- Functions continuous at specific x-values
- Removing discontinuities (factoring)
- Removing discontinuities (rationalization)
- Introduction to infinite limits
- Infinite limits and asymptotes
- Analyzing unbounded limits: rational function
- Analyzing unbounded limits: mixed function
- Introduction to limits at infinity
- Functions with same limit at infinity
- Limits at infinity of quotients (Part 1)
- Limits at infinity of quotients (Part 2)
- Limits at infinity of quotients with square roots (odd power)
- Limits at infinity of quotients with square roots (even power)
- Intermediate value theorem
- Worked example: using the intermediate value theorem
- Justification with the intermediate value theorem: table
- Justification with the intermediate value theorem: equation
- Formal definition of limits Part 1: intuition review
- Formal definition of limits Part 2: building the idea
- Formal definition of limits Part 3: the definition
- Formal definition of limits Part 4: using the definition
2. Differentiation: definition and basic derivative rules
46 فيديو تعليمي.
موضوعات ودروس الوحدة
- Newton, Leibniz, and Usain Bolt
- Derivative as a concept
- Secant lines & average rate of change
- Derivative as slope of curve
- The derivative & tangent line equations
- Formal definition of the derivative as a limit
- Formal and alternate form of the derivative
- Worked example: Derivative as a limit
- Worked example: Derivative from limit expression
- The derivative of x² at x=3 using the formal definition
- The derivative of x² at any point using the formal definition
- Estimating derivatives
- Differentiability and continuity
- Differentiability at a point: graphical
- Differentiability at a point: algebraic (function is differentiable)
- Differentiability at a point: algebraic (function isn't differentiable)
- Power rule
- Power rule (with rewriting the expression)
- Basic derivative rules
- Basic derivative rules: find the error
- Basic derivative rules: table
- Differentiating polynomials
- Differentiating integer powers (mixed positive and negative)
- Tangents of polynomials
- Derivatives of sin(x) and cos(x)
- Worked example: Derivatives of sin(x) and cos(x)
- Derivative of 𝑒ˣ
- Derivative of ln(x)
- Product rule
- Differentiating products
- Worked example: Product rule with table
- Worked example: Product rule with mixed implicit & explicit
- Quotient rule
- Worked example: Quotient rule with table
- Differentiating rational functions
- Derivatives of tan(x) and cot(x)
- Derivatives of sec(x) and csc(x)
- Proof: Differentiability implies continuity
- Justifying the power rule
- Proof of power rule for positive integer powers
- Proof of power rule for square root function
- Limit of sin(x)/x as x approaches 0
- Limit of (1-cos(x))/x as x approaches 0
- Proof of the derivative of sin(x)
- Proof of the derivative of cos(x)
- Product rule proof
3. Differentiation: composite, implicit, and inverse functions
38 فيديو تعليمي.
موضوعات ودروس الوحدة
- Chain rule
- Common chain rule misunderstandings
- Identifying composite functions
- Worked example: Derivative of cos³(x) using the chain rule
- Worked example: Derivative of √(3x²-x) using the chain rule
- Worked example: Derivative of ln(√x) using the chain rule
- Worked example: Chain rule with table
- Derivative of aˣ (for any positive base a)
- Derivative of logₐx (for any positive base a≠1)
- Worked example: Derivative of 7^(x²-x) using the chain rule
- Worked example: Derivative of log₄(x²+x) using the chain rule
- Worked example: Derivative of sec(3π/2-x) using the chain rule
- Worked example: Derivative of ∜(x³+4x²+7) using the chain rule
- Implicit differentiation
- Worked example: Implicit differentiation
- Worked example: Evaluating derivative with implicit differentiation
- Showing explicit and implicit differentiation give same result
- Derivatives of inverse functions
- Derivatives of inverse functions: from equation
- Derivatives of inverse functions: from table
- Derivative of inverse sine
- Derivative of inverse cosine
- Derivative of inverse tangent
- Differentiating functions: Find the error
- Manipulating functions before differentiation
- Differentiating using multiple rules: strategy
- Applying the chain rule and product rule
- Applying the chain rule twice
- Derivative of eᶜᵒˢˣ⋅cos(eˣ)
- Derivative of sin(ln(x²))
- Second derivatives
- Second derivatives (implicit equations): find expression
- Second derivatives (implicit equations): evaluate derivative
- Disguised derivatives
- Proof: Differentiability implies continuity
- If function u is continuous at x, then Δu→0 as Δx→0
- Chain rule proof
- Quotient rule from product & chain rules
4. Contextual applications of differentiation
26 فيديو تعليمي.
موضوعات ودروس الوحدة
- Interpreting the meaning of the derivative in context
- Introduction to one-dimensional motion with calculus
- Interpreting direction of motion from position-time graph
- Interpreting direction of motion from velocity-time graph
- Interpreting change in speed from velocity-time graph
- Worked example: Motion problems with derivatives
- Applied rate of change: forgetfulness
- Related rates intro
- Analyzing related rates problems: expressions
- Analyzing related rates problems: equations (Pythagoras)
- Analyzing related rates problems: equations (trig)
- Differentiating related functions intro
- Worked example: Differentiating related functions
- Related rates: Approaching cars
- Related rates: Falling ladder
- Related rates: water pouring into a cone
- Related rates: shadow
- Related rates: balloon
- Local linearity
- Local linearity and differentiability
- Worked example: Approximation with local linearity
- Linear approximation of a rational function
- L'Hôpital's rule introduction
- L'Hôpital's rule: limit at 0 example
- L'Hôpital's rule: limit at infinity example
- Proof of special case of l'Hôpital's rule
5. Applying derivatives to analyze functions
46 فيديو تعليمي.
موضوعات ودروس الوحدة
- Mean value theorem
- Mean value theorem example: polynomial
- Mean value theorem example: square root function
- Justification with the mean value theorem: table
- Justification with the mean value theorem: equation
- Mean value theorem application
- Extreme value theorem
- Critical points introduction
- Finding critical points
- Finding decreasing interval given the function
- Finding increasing interval given the derivative
- Introduction to minimum and maximum points
- Finding relative extrema (first derivative test)
- Worked example: finding relative extrema
- Analyzing mistakes when finding extrema (example 1)
- Analyzing mistakes when finding extrema (example 2)
- Finding absolute extrema on a closed interval
- Absolute minima & maxima (entire domain)
- Concavity introduction
- Analyzing concavity (graphical)
- Inflection points introduction
- Inflection points (graphical)
- Analyzing concavity (algebraic)
- Inflection points (algebraic)
- Mistakes when finding inflection points: second derivative undefined
- Mistakes when finding inflection points: not checking candidates
- Second derivative test
- Curve sketching with calculus: polynomial
- Curve sketching with calculus: logarithm
- Analyzing a function with its derivative
- Calculus-based justification for function increasing
- Justification using first derivative
- Inflection points from graphs of function & derivatives
- Justification using second derivative: inflection point
- Justification using second derivative: maximum point
- Connecting f, f', and f'' graphically
- Connecting f, f', and f'' graphically (another example)
- Optimization: sum of squares
- Optimization: box volume (Part 1)
- Optimization: box volume (Part 2)
- Optimization: profit
- Optimization: cost of materials
- Optimization: area of triangle & square (Part 1)
- Optimization: area of triangle & square (Part 2)
- Motion problems: finding the maximum acceleration
- Horizontal tangent to implicit curve
6. Integration and accumulation of change
69 فيديو تعليمي.
موضوعات ودروس الوحدة
- Introduction to integral calculus
- Definite integrals intro
- Worked example: accumulation of change
- Riemann approximation introduction
- Over- and under-estimation of Riemann sums
- Worked example: finding a Riemann sum using a table
- Worked example: over- and under-estimation of Riemann sums
- Midpoint sums
- Trapezoidal sums
- Summation notation
- Worked examples: Summation notation
- Riemann sums in summation notation
- Worked example: Riemann sums in summation notation
- Definite integral as the limit of a Riemann sum
- Worked example: Rewriting definite integral as limit of Riemann sum
- Worked example: Rewriting limit of Riemann sum as definite integral
- The fundamental theorem of calculus and accumulation functions
- Functions defined by definite integrals (accumulation functions)
- Finding derivative with fundamental theorem of calculus
- Finding derivative with fundamental theorem of calculus: chain rule
- Interpreting the behavior of accumulation functions
- Negative definite integrals
- Finding definite integrals using area formulas
- Definite integral over a single point
- Integrating scaled version of function
- Switching bounds of definite integral
- Integrating sums of functions
- Worked examples: Finding definite integrals using algebraic properties
- Definite integrals on adjacent intervals
- Worked example: Breaking up the integral's interval
- Worked example: Merging definite integrals over adjacent intervals
- Functions defined by integrals: switched interval
- Finding derivative with fundamental theorem of calculus: x is on lower bound
- Finding derivative with fundamental theorem of calculus: x is on both bounds
- The fundamental theorem of calculus and definite integrals
- Antiderivatives and indefinite integrals
- Reverse power rule
- Indefinite integrals: sums & multiples
- Rewriting before integrating
- Indefinite integral of 1/x
- Indefinite integrals of sin(x), cos(x), and eˣ
- Definite integrals: reverse power rule
- Definite integral of rational function
- Definite integral of radical function
- Definite integral of trig function
- Definite integral involving natural log
- Definite integral of piecewise function
- Definite integral of absolute value function
- 𝘶-substitution intro
- 𝘶-substitution: multiplying by a constant
- 𝘶-substitution: defining 𝘶
- 𝘶-substitution: defining 𝘶 (more examples)
- 𝘶-substitution: rational function
- 𝘶-substitution: logarithmic function
- 𝘶-substitution: definite integrals
- 𝘶-substitution: definite integral of exponential function
- Integration using long division
- Integration using completing the square and the derivative of arctan(x)
- Integration by parts intro
- Integration by parts: ∫x⋅cos(x)dx
- Integration by parts: ∫ln(x)dx
- Integration by parts: ∫x²⋅𝑒ˣdx
- Integration by parts: ∫𝑒ˣ⋅cos(x)dx
- Integration by parts: definite integrals
- Integration with partial fractions
- Introduction to improper integrals
- Divergent improper integral
- Proof of fundamental theorem of calculus
- Intuition for second part of fundamental theorem of calculus
7. Differential equations
27 فيديو تعليمي.
موضوعات ودروس الوحدة
- Differential equations introduction
- Writing a differential equation
- Verifying solutions to differential equations
- Slope fields introduction
- Worked example: equation from slope field
- Worked example: slope field from equation
- Worked example: forming a slope field
- Approximating solution curves in slope fields
- Worked example: range of solution curve from slope field
- Euler's method
- Worked example: Euler's method
- Separable equations introduction
- Addressing treating differentials algebraically
- Worked example: separable differential equations
- Worked example: identifying separable equations
- Particular solutions to differential equations: rational function
- Particular solutions to differential equations: exponential function
- Worked example: finding a specific solution to a separable equation
- Worked example: separable equation with an implicit solution
- Exponential models & differential equations (Part 1)
- Exponential models & differential equations (Part 2)
- Worked example: exponential solution to differential equation
- Growth models: introduction
- The logistic growth model
- Worked example: Logistic model word problem
- Logistic equations (Part 1)
- Logistic equations (Part 2)
8. Applications of integration
39 فيديو تعليمي.
موضوعات ودروس الوحدة
- Average value over a closed interval
- Calculating average value of function over interval
- Mean value theorem for integrals
- Motion problems with integrals: displacement vs. distance
- Analyzing motion problems: position
- Analyzing motion problems: total distance traveled
- Worked example: motion problems (with definite integrals)
- Average acceleration over interval
- Area under rate function gives the net change
- Interpreting definite integral as net change
- Worked examples: interpreting definite integrals in context
- Analyzing problems involving definite integrals
- Worked example: problem involving definite integral (algebraic)
- Area between a curve and the x-axis
- Area between a curve and the x-axis: negative area
- Area between curves
- Worked example: area between curves
- Composite area between curves
- Area between a curve and the 𝘺-axis
- Horizontal area between curves
- Volume with cross sections: intro
- Volume with cross sections: squares and rectangles (no graph)
- Volume with cross sections perpendicular to y-axis
- Volume with cross sections: semicircle
- Volume with cross sections: triangle
- Disc method around x-axis
- Generalizing disc method around x-axis
- Disc method around y-axis
- Disc method rotation around horizontal line
- Disc method rotating around vertical line
- Calculating integral disc around vertical line
- Solid of revolution between two functions (leading up to the washer method)
- Generalizing the washer method
- Washer method rotating around horizontal line (not x-axis), part 1
- Washer method rotating around horizontal line (not x-axis), part 2
- Washer method rotating around vertical line (not y-axis), part 1
- Washer method rotating around vertical line (not y-axis), part 2
- Arc length intro
- Worked example: arc length
9. Parametric equations, polar coordinates, and vector-valued functions
18 فيديو تعليمي.
موضوعات ودروس الوحدة
- Parametric equations intro
- Parametric equations differentiation
- Second derivatives (parametric functions)
- Parametric curve arc length
- Worked example: Parametric arc length
- Vector-valued functions intro
- Vector-valued functions differentiation
- Second derivatives (vector-valued functions)
- Planar motion example: acceleration vector
- Motion along a curve: finding rate of change
- Motion along a curve: finding velocity magnitude
- Planar motion (with integrals)
- Polar functions derivatives
- Worked example: differentiating polar functions
- Area bounded by polar curves
- Worked example: Area enclosed by cardioid
- Worked example: Area between two polar graphs
- Evaluating definite integral with calculator
10. Infinite sequences and series
61 فيديو تعليمي.
موضوعات ودروس الوحدة
- Convergent and divergent sequences
- Worked example: sequence convergence/divergence
- Partial sums intro
- Partial sums: formula for nth term from partial sum
- Partial sums: term value from partial sum
- Infinite series as limit of partial sums
- Worked example: convergent geometric series
- Worked example: divergent geometric series
- Infinite geometric series word problem: bouncing ball
- Infinite geometric series word problem: repeating decimal
- nth term divergence test
- Integral test
- Worked example: Integral test
- Harmonic series and 𝑝-series
- Worked example: p-series
- Direct comparison test
- Worked example: direct comparison test
- Limit comparison test
- Worked example: limit comparison test
- Proof: harmonic series diverges
- Alternating series test
- Worked example: alternating series
- Ratio test
- Conditional & absolute convergence
- Alternating series remainder
- Worked example: alternating series remainder
- Taylor & Maclaurin polynomials intro (part 1)
- Taylor & Maclaurin polynomials intro (part 2)
- Worked example: Maclaurin polynomial
- Worked example: coefficient in Maclaurin polynomial
- Worked example: coefficient in Taylor polynomial
- Visualizing Taylor polynomial approximations
- Taylor polynomial remainder (part 1)
- Taylor polynomial remainder (part 2)
- Worked example: estimating sin(0.4) using Lagrange error bound
- Worked example: estimating eˣ using Lagrange error bound
- Power series intro
- Worked example: interval of convergence
- Function as a geometric series
- Geometric series as a function
- Power series of arctan(2x)
- Power series of ln(1+x³)
- Maclaurin series of cos(x)
- Maclaurin series of sin(x)
- Maclaurin series of eˣ
- Worked example: power series from cos(x)
- Worked example: cosine function from power series
- Worked example: recognizing function from Taylor series
- Visualizing Taylor series approximations
- Euler's formula & Euler's identity
- Geometric series interval of convergence
- Integrating power series
- Differentiating power series
- Finding function from power series by integrating
- Interval of convergence for derivative and integral
- Formal definition for limit of a sequence
- Proving a sequence converges using the formal definition
- Finite geometric series formula
- Infinite geometric series formula intuition
- Proof of infinite geometric series as a limit
- Proof of p-series convergence criteria
11. AP Calculus BC solved exams
30 فيديو تعليمي.
موضوعات ودروس الوحدة
- 2017 AP Calculus AB/BC 4a
- 2017 AP Calculus AB/BC 4b
- 2017 AP Calculus AB/BC 4c
- 2015 AP Calculus BC 2a
- 2015 AP Calculus BC 2b
- 2015 AP Calculus BC 2c
- 2015 AP Calculus BC 2d
- 2015 AP Calculus BC 5a
- 2015 AP Calculus BC 5b
- 2015 AP Calculus BC 5c
- 2015 AP Calculus BC 5d
- 2015 AP Calculus BC 6a
- 2015 AP Calculus BC 6b
- 2015 AP Calculus BC 6c
- 2011 Calculus BC free response #1a
- 2011 Calculus BC free response #1 (b & c)
- 2011 Calculus BC free response #1d
- 2011 Calculus BC free response #3a
- 2011 Calculus BC free response #3 (b & c)
- 2011 Calculus BC free response #6a
- 2011 Calculus BC free response #6b
- 2011 Calculus BC free response #6c
- 2011 Calculus BC free response #6d
- AP Calculus BC exams: 2008 1 a
- AP Calculus BC exams: 2008 1 b&c
- AP Calculus BC exams: 2008 1 c&d
- AP Calculus BC exams: 2008 1 d
- Calculus BC 2008 2 a
- Calculus BC 2008 2 b &c
- Calculus BC 2008 2d
مقررات مرتبطة
- حساب المثلثات | Trigonometry
- ما قبل التفاضل والتكامل | Precalculus
- الإحصاء والاحتمالات | Statistics & probability
- التفاضل والتكامل AP (AB) | AP®︎ Calculus AB
- الإحصاء AP | AP®︎ Statistics
- التفاضل والتكامل متعدد المتغيرات | Multivariable calculus
المصدر والحقوق
المحتوى الأصلي المرخص: أكاديمية خان. تقوم مِسبار بالفهرسة والتنظيم حسب التخصص، وتعرض الفيديوهات المرخصة غير التجارية داخل مكتبة مجانية مستقلة.