Calculus Overview
Calculus is the mathematics of change and motion. This comprehensive course covers differential and integral calculus, preparing students for AP Calculus AB/BC exams and college-level mathematics. Students develop both computational skills and conceptual understanding.
- Master limits and continuity concepts
- Understand and apply derivatives to analyze functions and solve problems
- Master integration techniques and applications
- Work with sequences, series, and differential equations
Calculus Standards
Limits and Continuity
Standard 1: Limits - Definition and Interpretation
Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence.
- 1.1: Prove and use theorems evaluating limits of sums, products, quotients, and composition of functions
- 1.2: Use graphical calculators to verify and estimate limits
- 1.3: Prove and use special limits, such as lim(sin(x)/x) = 1 and lim((1-cos(x))/x) = 0 as x → 0
lim(x→2) (x² - 4)/(x - 2) = lim(x→2) (x + 2) = 4
lim(x→0) sin(x)/x = 1 (special limit)
Standard 2: Continuity
Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. A function is continuous at x = a if lim(x→a) f(x) = f(a).
f(x) = x² is continuous everywhere because lim(x→a) x² = a² = f(a) for all a
f(x) = 1/x is not continuous at x = 0 because f(0) is undefined
Standard 3: Intermediate and Extreme Value Theorems
Students demonstrate an understanding and the application of the intermediate value theorem (if f is continuous on [a,b], it takes on all values between f(a) and f(b)) and the extreme value theorem (continuous functions on closed intervals attain maximum and minimum values).
If f(1) = -2 and f(3) = 5, and f is continuous, then f(c) = 0 for some c in (1,3) by IVT
f(x) = x² on [0,3] has minimum value 0 at x=0 and maximum value 9 at x=3 by EVT
Derivatives
Standard 4: Definition and Interpretation of Derivatives
Students demonstrate understanding of the formal definition of the derivative of a function at a point and the notion of differentiability: f'(a) = lim(h→0) [f(a+h) - f(a)]/h
- 4.1: Understand derivative as slope of tangent line
- 4.2: Interpret derivative as instantaneous rate of change; apply to physics, chemistry, economics
- 4.3: Understand relation between differentiability and continuity (differentiable implies continuous)
- 4.4: Derive formulas for derivatives of algebraic, trig, inverse trig, exponential, and logarithmic functions
Find derivative: f(x) = x³ → f'(x) = 3x²
Using definition: f'(2) = lim(h→0) [(2+h)³ - 8]/h = 12
Standard 5: Chain Rule
Students know the chain rule and its proof and applications: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x)
Chain rule: d/dx[sin(x²)] = cos(x²) · 2x
d/dx[(3x + 1)⁵] = 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴
Standard 6: Parametric and Implicit Differentiation
Students find derivatives of parametrically defined functions and use implicit differentiation in various applications.
Implicit: x² + y² = 25 → 2x + 2y(dy/dx) = 0 → dy/dx = -x/y
Parametric: x = t², y = t³ → dy/dx = (dy/dt)/(dx/dt) = 3t²/2t = 3t/2
Standard 7: Higher Order Derivatives
Students compute derivatives of higher orders: f''(x), f'''(x), etc.
f(x) = x⁴ → f'(x) = 4x³ → f''(x) = 12x² → f'''(x) = 24x
For position s(t), velocity v = s'(t), acceleration a = s''(t) = v'(t)
Standard 8: Mean Value Theorem
Students know and can apply Rolle's Theorem, the Mean Value Theorem (if f is continuous on [a,b] and differentiable on (a,b), then f'(c) = [f(b)-f(a)]/(b-a) for some c), and L'Hôpital's rule for indeterminate forms.
MVT: For f(x) = x² on [1,3], f'(c) = (9-1)/(3-1) = 4, so 2c = 4 → c = 2
L'Hôpital's: lim(x→0) sin(x)/x = lim(x→0) cos(x)/1 = 1
Standard 9: Curve Sketching
Students use differentiation to sketch graphs of functions by hand, identifying maxima, minima, inflection points, and intervals where the function is increasing/decreasing.
f(x) = x³ - 3x: f'(x) = 3x² - 3 = 0 at x = ±1
f''(x) = 6x: concave down for x < 0, concave up for x > 0
Local max at x = -1, local min at x = 1, inflection point at x = 0
Standard 10: Newton's Method
Students know Newton's method for approximating zeros: x_(n+1) = x_n - f(x_n)/f'(x_n)
To find √2, solve x² - 2 = 0: x_(n+1) = x_n - (x_n² - 2)/(2x_n)
Starting with x₀ = 1: x₁ = 1.5, x₂ ≈ 1.4167, x₃ ≈ 1.4142 ≈ √2
Standard 11: Optimization
Students use differentiation to solve optimization (maximum-minimum) problems in pure and applied contexts.
Max area of rectangle with perimeter 100: Let width = x, length = 50 - x
Area A(x) = x(50 - x) = 50x - x². A'(x) = 50 - 2x = 0 → x = 25
Maximum area = 25×25 = 625
Standard 12: Related Rates
Students use differentiation to solve related rate problems in various contexts.
Ladder sliding down wall: x² + y² = 25 (ladder length 5m)
Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0
If dx/dt = 2 m/s when x = 3, y = 4, find dy/dt: 6(2) + 8(dy/dt) = 0 → dy/dt = -1.5 m/s
Integrals
Standard 13: Definite Integral Definition
Students know the definition of the definite integral by using Riemann sums and use this definition to approximate integrals: ∫f(x)dx = lim(n→∞) Σf(x_i)Δx
∫₀¹ x²dx ≈ Σf(x_i)Δx with Δx = 1/n, x_i = i/n
As n → ∞, Riemann sum converges to exact value 1/3
Standard 14: Integral Applications
Students apply the definition of the integral to model problems in physics, economics, obtaining results in terms of integrals.
Total distance = ∫ₐᵇ v(t)dt where v(t) is velocity
Total revenue from marginal revenue: ∫ₐᵇ MR(x)dx
Standard 15: Fundamental Theorem of Calculus
Students demonstrate knowledge and proof of the fundamental theorem: (1) d/dx[∫ₐˣ f(t)dt] = f(x), and (2) ∫ₐᵇ f(x)dx = F(b) - F(a) where F' = f
∫₀¹ x²dx = [x³/3]₀¹ = 1/3 - 0 = 1/3
d/dx[∫₀ˣ t²dt] = x²
Standard 16: Applications of Integrals
Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of surface of revolution, length of a curve, and work.
Volume of solid of revolution: V = ∫ₐᵇ π[f(x)]²dx
Arc length: L = ∫ₐᵇ √[1 + (dy/dx)²]dx
Work: W = ∫ₐᵇ F(x)dx
Standard 17: Integration Techniques - Basic
Students compute integrals by hand using techniques such as substitution, integration by parts, and trigonometric substitution. They can combine these techniques when appropriate.
Substitution: ∫ 2x·e^(x²)dx, let u = x², du = 2xdx → ∫ e^u du = e^(x²) + C
Integration by parts: ∫ x·e^x dx = xe^x - e^x + C
Standard 18: Inverse Trigonometric Functions
Students know definitions and properties of inverse trig functions and express these functions as indefinite integrals.
∫ 1/(1+x²)dx = arctan(x) + C
∫ 1/√(1-x²)dx = arcsin(x) + C
Standard 19: Rational Function Integration
Students compute integrals of rational functions by combining techniques with algebraic methods of partial fractions and completing the square.
∫ 1/(x²-1)dx = ∫ [1/2(x-1) - 1/2(x+1)]dx = (1/2)ln|x-1| - (1/2)ln|x+1| + C
Standard 20: Trigonometric Integration
Students compute integrals of trigonometric functions using various techniques.
∫ sin²(x)dx = ∫ (1-cos(2x))/2 dx = x/2 - sin(2x)/4 + C
∫ tan(x)dx = -ln|cos(x)| + C
Standard 21: Numerical Integration
Students understand algorithms for Simpson's rule and Newton's method. They use calculators or computers to approximate integrals numerically.
Trapezoidal rule: ∫ₐᵇ f(x)dx ≈ (b-a)/2n · [f(x₀) + 2f(x₁) + ... + 2f(x_(n-1)) + f(xₙ)]
Simpson's rule gives more accurate approximations using parabolic arcs
Standard 22: Improper Integrals
Students understand improper integrals as limits of definite integrals: ∫ₐ^∞ f(x)dx = lim(b→∞) ∫ₐᵇ f(x)dx
∫₁^∞ 1/x²dx = lim(b→∞) ∫₁ᵇ 1/x²dx = lim(b→∞) [-1/x]₁ᵇ = lim(b→∞) (1 - 1/b) = 1
∫₁^∞ 1/x dx diverges (approaches ∞)
Series and Differential Equations
Standard 23: Convergence and Divergence of Series
Students demonstrate understanding of definitions of convergence and divergence of sequences and series of real numbers. By using tests such as comparison test, ratio test, and alternating series test, they determine whether a series converges.
Geometric series: Σ(1/2)ⁿ = 1/(1-1/2) = 2 (converges)
Harmonic series: Σ(1/n) diverges
Ratio test: For Σaₙ, if lim|aₙ₊₁/aₙ| < 1, series converges
Standard 24: Radius of Convergence
Students understand and can compute the radius (interval) of convergence of power series using ratio test or other methods.
Power series: Σxⁿ/n! = e^x, radius of convergence R = ∞
For Σxⁿ, R = 1 (converges for |x| < 1)
Standard 25: Term-by-Term Operations
Students differentiate and integrate the terms of a power series to form new series from known ones.
If Σxⁿ = 1/(1-x) for |x| < 1
Then d/dx[Σxⁿ] = Σnxⁿ⁻¹ = 1/(1-x)²
And ∫[Σxⁿ]dx = Σxⁿ⁺¹/(n+1) = -ln(1-x)
Standard 26: Taylor Series
Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term. Taylor series: f(x) = Σ[f^(n)(a)/n!]·(x-a)ⁿ
Taylor series for e^x at a=0: e^x = 1 + x + x²/2! + x³/3! + ...
sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...
cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...
Standard 27: Differential Equations
Students know techniques of solution of selected elementary differential equations and their applications to growth-and-decay problems, separable equations, and other modeling situations.
Exponential growth/decay: dy/dx = ky → y = Ce^(kx)
Separable: dy/dx = xy → ∫(1/y)dy = ∫xdx → ln|y| = x²/2 + C → y = Ae^(x²/2)
Population model: dP/dt = kP → P(t) = P₀e^(kt)