Calculus I
Calculus I covers single-variable differential and integral calculus from limits and continuity through derivatives and their applications (optimization, related rates, curve sketching) to the Fundamental Theorem of Calculus, u-substitution, area between curves, volumes of revolution, and separable differential equations.
Who Should Take This
Designed for students entering college calculus, advanced high school students taking AP Calculus AB, and STEM professionals who need to refresh calculus fundamentals. Pre-Algebra and Pre-Calculus proficiency (including trigonometry and exponential/logarithmic functions) is required before starting.
What's Covered
1Limits and Continuity
2Derivatives and Differentiation Rules
3Applications of Derivatives
4Integration
5Geometric Applications of Integration
6Introduction to Differential Equations
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
13 Activity Formats
Course Outline
1Limits and Continuity 9 topics
Describe the intuitive meaning of lim(x→a) f(x) = L as the value f(x) approaches as x gets arbitrarily close to a from both sides, without requiring f(a) to equal L
Apply limit laws (sum, difference, product, quotient, constant multiple, root) to evaluate limits of algebraic functions by direct substitution when f is continuous at the point
Apply algebraic techniques (factoring, rationalizing, common denominator) to evaluate indeterminate forms 0/0 after canceling the factor causing the zero in the denominator
Apply one-sided limit evaluation and use the result that a two-sided limit exists only when left-hand and right-hand limits both exist and are equal, identifying jump discontinuities from one-sided limit discrepancy
Apply limits at infinity to determine horizontal asymptotes by evaluating lim(x→±∞) f(x), using the dominance hierarchy of functions (polynomial vs rational vs exponential) and L'Hôpital's rule for ∞/∞ forms
Describe continuity at a point requiring f(a) to be defined, lim(x→a) f(x) to exist, and the limit to equal f(a), classify removable vs jump vs infinite discontinuities, and apply the Intermediate Value Theorem
Analyze the Squeeze Theorem to evaluate limits like lim(x→0) x²sin(1/x) = 0 by bounding the function between two simpler functions that converge to the same limit
Apply the limit definition of the derivative to recognize the difference quotient pattern in non-standard forms and connect it to instantaneous rate of change in position, temperature, and cost function contexts
Apply the special trigonometric limits lim(x→0) sin(x)/x = 1 and lim(x→0) (1-cos x)/x = 0 to evaluate composite limits involving sinusoidal expressions and explain these as the foundation for the derivative of sin x
2Derivatives and Differentiation Rules 11 topics
Describe the derivative f'(a) as the limit of the difference quotient [f(a+h)-f(a)]/h as h→0, interpret it as the instantaneous rate of change, and compute derivatives from the definition for simple polynomial and rational functions
Apply the power rule d/dx[xⁿ] = nxⁿ⁻¹, constant rule, sum and difference rules, and constant multiple rule to differentiate polynomial and radical expressions efficiently without the limit definition
Apply the product rule d/dx[uv] = u'v + uv' and quotient rule d/dx[u/v] = (u'v - uv')/v² to differentiate products and quotients of functions, avoiding common sign errors in the quotient rule
Apply the chain rule d/dx[f(g(x))] = f'(g(x))·g'(x) to differentiate composite functions including power chains, trig of polynomials, and nested compositions requiring multiple applications
Apply derivatives of trigonometric functions (d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[tan x] = sec²x, and the remaining three) combined with the chain rule for composite trig expressions
Apply derivatives of exponential and natural logarithm (d/dx[eˣ] = eˣ, d/dx[ln x] = 1/x, d/dx[aˣ] = aˣ ln a, d/dx[logₐx] = 1/(x ln a)) and logarithmic differentiation for products of many functions
Apply implicit differentiation by differentiating both sides of an equation with respect to x using the chain rule for y terms, then solving for dy/dx to find slopes of implicitly defined curves
Analyze higher-order derivatives by computing f'', f''', and noting physical interpretations where f' is velocity, f'' is acceleration, and f'' positive means concave up when f represents position
Apply logarithmic differentiation to differentiate y = [f(x)]^g(x) (variable in both base and exponent) by taking ln of both sides, differentiating implicitly, and solving for dy/dx
Apply derivatives of inverse trigonometric functions d/dx[arcsin x] = 1/√(1-x²), d/dx[arctan x] = 1/(1+x²), and d/dx[arcsec x] = 1/(|x|√(x²-1)) to compute derivatives of expressions containing inverse trig functions
Describe the geometric interpretation of the derivative as the slope of the tangent line to f at the point (a, f(a)), write the equation of the tangent line using y - f(a) = f'(a)(x - a), and identify the normal line as perpendicular to it
3Applications of Derivatives 11 topics
Apply the first derivative test to find critical points where f'(x) = 0 or undefined, classify each as local max, local min, or neither by analyzing the sign change of f' across the critical point
Apply concavity analysis using f'' to identify intervals of concave up (f'' > 0) and concave down (f'' < 0), locate inflection points where f'' changes sign, and use the second derivative test to classify critical points
Apply the closed interval method to find absolute extrema on [a, b] by evaluating f at all critical points in (a, b) and at both endpoints, then comparing all values to identify the global maximum and minimum
Apply optimization problem-solving by translating a word problem into an objective function with a constraint, using the constraint to reduce to one variable, differentiating, and verifying the critical point is an extremum
Apply related rates by differentiating an equation relating several variables with respect to time using the chain rule, substituting known rates and values, and solving for the unknown rate
Apply L'Hôpital's rule to evaluate indeterminate forms 0/0 and ∞/∞ by differentiating numerator and denominator separately, and apply it iteratively when the result remains indeterminate
Analyze curve sketching by combining domain, intercepts, asymptotes, intervals of increase/decrease, local extrema, concavity, and inflection points to produce an accurate qualitative sketch of a function
Apply the Mean Value Theorem to guarantee a point c in (a, b) where f'(c) equals the average rate of change, verify the theorem's hypotheses (continuous on [a,b], differentiable on (a,b)), and apply Rolle's Theorem as the special case
Describe Newton's Method for approximating roots of f(x) = 0 as the iteration xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ), perform two or three iterations from an initial guess, and explain geometrically that each step follows the tangent line to find the next x-intercept
Apply linearization L(x) = f(a) + f'(a)(x - a) as a linear approximation to f near x = a, use it to estimate function values like √(8.9) ≈ √9 + correction, and compute the differential dy = f'(x)dx to estimate changes in function output
Analyze the relationship between a function being differentiable and being continuous, and explain with an example why continuity at a point does not guarantee differentiability (e.g., |x| at 0 is continuous but has a corner, not a derivative)
4Integration 12 topics
Describe antiderivatives as functions F such that F'(x) = f(x), state the general antiderivative F(x) + C with the constant of integration, and apply the power rule for integration ∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ -1
Apply integration rules for standard functions including ∫eˣdx = eˣ+C, ∫(1/x)dx = ln|x|+C, ∫cos x dx = sin x+C, ∫sin x dx = -cos x+C, and ∫sec²x dx = tan x+C
Describe Riemann sums as approximations to the area under a curve using left, right, or midpoint rectangles of width Δx, and explain how the definite integral ∫ₐᵇf(x)dx is the limit of Riemann sums as Δx → 0
Apply the Fundamental Theorem of Calculus Part I: d/dx[∫ₐˣf(t)dt] = f(x), explaining that differentiation and integration are inverse operations and evaluating derivatives of integral-defined functions
Apply the Fundamental Theorem of Calculus Part II: ∫ₐᵇf(x)dx = F(b) - F(a) where F' = f, compute definite integrals of polynomial, trigonometric, and exponential functions using antiderivatives
Apply u-substitution to evaluate integrals of composite functions by choosing u = g(x), computing du = g'(x)dx, transforming the integral in terms of u, integrating, and back-substituting
Apply definite integral properties including linearity, additivity over intervals, comparison properties, and the interpretation of ∫ₐᵇf(x)dx as net signed area (negative when f < 0 below the x-axis)
Analyze the net signed area vs total area distinction by identifying where f(x) < 0 and splitting integrals at x-intercepts to compute total area as the sum of absolute values of signed areas
Apply definite integral approximations using left, right, and midpoint Riemann sums with n rectangles and compare the accuracy improvement as n increases, explaining the error bound for each method when f is monotone
Apply the trapezoidal rule ∫ₐᵇf(x)dx ≈ (Δx/2)[f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)] to approximate definite integrals, and explain why the trapezoid rule is more accurate than rectangle rules for smooth functions
Analyze the relationship between position, velocity, and acceleration using integration — given velocity v(t), compute position displacement ∫ₐᵇv(t)dt and total distance ∫ₐᵇ|v(t)|dt by splitting at t values where v = 0
Apply u-substitution to definite integrals by either changing the limits of integration to match the new variable u or back-substituting after integration, and verify that both approaches produce the same numerical result
5Geometric Applications of Integration 8 topics
Apply the area between two curves formula ∫ₐᵇ[f(x) - g(x)]dx where f(x) ≥ g(x) on [a, b], and identify intersection points by solving f(x) = g(x) to determine the limits of integration
Apply area between curves with respect to y by integrating horizontally when the region is more naturally described with x as a function of y, setting up ∫ₓ₁ˣ²[right - left]dy
Apply the disk method to compute volumes of solids of revolution by integrating π[f(x)]² over the interval when rotating around the x-axis, and π[f(y)]² when rotating around the y-axis
Apply the washer method to compute volumes of revolution with a hollow region by integrating π([f(x)]² - [g(x)]²) when the region between two curves is rotated around an axis
Analyze which method (disk vs washer, x-integration vs y-integration) minimizes algebraic complexity for a given solid of revolution by comparing the geometry of the region and the axis of rotation
Apply average value of a function formula f_avg = (1/(b-a))∫ₐᵇf(x)dx to compute the average temperature, average velocity, or average force over an interval, and interpret it as the height of a rectangle with equal area
Analyze arc length calculations using ∫ₐᵇ√(1 + [f'(x)]²)dx at a conceptual level by setting up the integral for simple curves and interpreting each infinitesimal arc length element ds as the hypotenuse of an infinitesimal right triangle
Apply the shell method to compute volumes of revolution by integrating 2π ∫ₐᵇ x·f(x)dx when rotating around the y-axis, and identify when the shell method is simpler than the disk/washer method
6Introduction to Differential Equations 9 topics
Describe a differential equation as an equation relating a function to its derivative(s), identify the order of an ODE, and explain that a general solution contains an arbitrary constant C representing a family of solutions
Apply separation of variables to solve first-order separable ODEs by rearranging dy/dx = f(x)g(y) into (1/g(y))dy = f(x)dx, integrating both sides, and applying initial conditions to find particular solutions
Apply exponential growth and decay models by solving dy/dt = ky to get y = y₀eᵏᵗ, identifying k > 0 as growth and k < 0 as decay, and computing half-life, doubling time, and population projections
Apply slope field interpretation by reading the direction at any point (x, y) from the differential equation's right-hand side, sketching representative solution curves, and predicting long-run behavior without solving analytically
Analyze connections between the original function, its derivative graph, and the antiderivative graph by identifying where f' = 0 matches extrema of f, and where f is increasing corresponds to f' > 0
Apply initial value problem solving by using the given initial condition to determine the constant C in the general antiderivative, and write the particular solution satisfying both the differential equation and the initial condition
Describe Newton's law of cooling as dT/dt = k(T - Tₑ) where Tₑ is the ambient temperature, solve the separable ODE, and apply it to compute when a cooling body reaches a target temperature
Apply the logistic growth model dP/dt = rP(1 - P/K) qualitatively by sketching the solution curve, identifying the inflection point at P = K/2 where growth is fastest, and explaining why population growth slows as it approaches carrying capacity K
Analyze partial derivatives conceptually by explaining that ∂f/∂x treats y as a constant, and interpreting partial derivatives as the rate of change in one direction on a surface — setting the stage for multivariable calculus without formal computation
Scope
Included Topics
- Limits and limit laws, one-sided limits, limits at infinity, continuity and the Intermediate Value Theorem, instantaneous rate of change and the derivative as a limit definition, derivative rules (constant, power, sum/difference, product, quotient, chain rules), derivatives of trigonometric functions (sin, cos, tan, sec, csc, cot), derivatives of exponential and logarithmic functions, higher-order derivatives, implicit differentiation, related rates, linearization and differentials, increasing/decreasing functions and the first derivative test, concavity and the second derivative test, extrema (absolute and local), optimization problems, rolle's theorem and the mean value theorem, L'Hôpital's rule, antiderivatives and indefinite integrals, Riemann sums and the definite integral, the Fundamental Theorem of Calculus (both parts), integration by substitution (u-substitution), area between curves, volume of revolution (disk and washer methods), introduction to differential equations (separable equations)
Not Covered
- Integration by parts, trigonometric substitution, partial fractions (Calculus II)
- Multivariable and vector calculus (Calculus III)
- Formal epsilon-delta proofs of limits (real analysis level)
- Infinite series and convergence tests (Calculus II)
- Differential equations beyond separable ODEs
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