Overview
This lecture covers essential calculus concepts, including rational expressions, limits and continuity, derivatives, the chain rule, related rates, integration, and the fundamental theorem of calculus. It emphasizes practical problem-solving strategies and introduces key techniques for evaluating functions and their rates of change, with a focus on understanding, computation, and application.
Rational Expressions
- A rational expression is a fraction where both the numerator and denominator are polynomials.
- To simplify a rational expression:
- Factor both the numerator and denominator completely.
- Cancel any common factors that appear in both.
- The simplified form should have no common factors between numerator and denominator.
- Multiplying rational expressions:
- Multiply the numerators together and the denominators together.
- Factor and simplify if possible before multiplying to make cancellation easier.
- Dividing rational expressions:
- Multiply by the reciprocal of the divisor (flip and multiply).
- Factor and cancel common factors before multiplying.
- Adding or subtracting rational expressions:
- Find the least common denominator (LCD) by factoring denominators and combining all necessary factors.
- Rewrite each fraction with the LCD as the denominator.
- Combine the numerators, then simplify the result.
- When solving rational equations:
- Multiply both sides by the LCD to clear denominators.
- Solve the resulting equation, but always check for extraneous solutions—values that make any denominator zero are not valid.
Limits & Continuity
- The limit as x approaches a value describes the behavior of a function near that value, not necessarily at the value itself.
- Limits may fail to exist due to:
- Jump discontinuities (the function "jumps" at a point).
- Vertical asymptotes (infinite limits).
- Wild oscillations (function does not settle to a value).
- Limit laws allow direct substitution if the function is continuous and the denominator does not become zero.
- A function is continuous at x = a if:
- The limit as x approaches a exists.
- The function is defined at a.
- The limit equals the function value at a.
- A function is continuous on an interval if it is continuous at every point in that interval. For closed intervals, check one-sided continuity at endpoints.
- Types of discontinuities:
- Jump: function jumps from one value to another.
- Removable (hole): function is undefined at a point but can be "plugged."
- Infinite: function approaches infinity (vertical asymptote).
- Wild: function oscillates without settling.
- The Intermediate Value Theorem:
- If a function is continuous on [a, b], it takes on every value between f(a) and f(b).
- Useful for proving the existence of roots or values within an interval.
Derivatives & Rates of Change
- The derivative at a point gives the slope of the tangent line and represents the instantaneous rate of change.
- The difference quotient, [f(x + h) – f(x)] / h, gives the average rate of change; as h → 0, it gives the instantaneous rate (the derivative).
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹; the derivative of a constant is 0.
- The derivative of eˣ is eˣ; the derivative of aˣ is ln(a)·aˣ.
- Higher-order derivatives (second, third, etc.) measure rates of change of rates (e.g., acceleration is the second derivative of position).
- The derivative can be interpreted as velocity (rate of change of position) or as the slope of a function at a point.
- The derivative of a sum or difference is the sum or difference of the derivatives.
- The product rule: d/dx[f(x)·g(x)] = f(x)·g'(x) + f'(x)·g(x).
- The quotient rule: d/dx[f(x)/g(x)] = (f'(x)·g(x) – f(x)·g'(x)) / [g(x)]².
Chain Rule & Implicit Differentiation
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x). Use for composite functions.
- For implicit differentiation:
- Differentiate both sides of an equation with respect to x, treating y as a function of x (use dy/dx when differentiating y terms).
- Useful for finding derivatives when y is not isolated or when dealing with curves not defined as functions.
- After differentiating, solve for dy/dx to find the derivative.
- Logarithmic differentiation:
- Take the natural log of both sides to simplify differentiation, especially when variables appear in both the base and exponent.
- Differentiate using implicit differentiation, then solve for dy/dx.
Related Rates & Applications
- Related rates problems involve finding how rates of change of different quantities are connected.
- Steps for solving related rates problems:
- Draw a diagram and label all variables.
- Assign variables to all quantities that change with time.
- Write equations relating the variables.
- Differentiate both sides with respect to time (usually t).
- Substitute known values and solve for the desired rate.
- Common applications include geometry (area, volume, distance), physics (motion, velocity, acceleration), and trigonometry (angles, rotation).
Integrals & Antiderivatives
- Integration finds the accumulated area under a curve; the antiderivative reverses differentiation.
- The general antiderivative: ∫f(x)dx = F(x) + C, where C is the constant of integration.
- Use substitution (u-substitution) to simplify integrals, especially when the integrand contains a function and its derivative.
- Definite integrals compute the net area between a function and the x-axis over an interval [a, b]; the result is a number.
- The average value of a function on [a, b] is (1/(b–a)) ∫ₐᵇ f(x) dx.
- The mean value theorem for integrals guarantees that a continuous function attains its average value at some point in the interval.
- Riemann sums approximate the area under a curve by summing the areas of rectangles; as the number of rectangles increases, the sum approaches the definite integral.
- The substitution method (u-substitution) is based on the chain rule and is used to evaluate integrals of composite functions.
Fundamental Theorem of Calculus
- Part 1: If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x). The derivative of the accumulated area function is the original function.
- Part 2: The definite integral from a to b of f(x) dx equals F(b) – F(a), where F is any antiderivative of f.
- This theorem connects differentiation and integration, allowing computation of definite integrals using antiderivatives.
- The mean value theorem for integrals: For a continuous function on [a, b], there exists c in [a, b] such that f(c) equals the average value of f on [a, b].
Maximum/Minimum Values
- A function has a local or absolute maximum/minimum at points where the derivative is zero or undefined (critical numbers).
- Critical numbers: values where f'(x) = 0 or f'(x) does not exist.
- First derivative test:
- If f' changes from positive to negative at a critical number, there is a local maximum.
- If f' changes from negative to positive, there is a local minimum.
- If f' does not change sign, there is no local extremum.
- Second derivative test:
- If f''(x) > 0 at a critical number, it's a local minimum (concave up).
- If f''(x) < 0, it's a local maximum (concave down).
- If f''(x) = 0, the test is inconclusive.
- Inflection points occur where the concavity changes (f'' changes sign).
- The Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), there is some c in (a, b) where f'(c) equals the average rate of change over [a, b].
Key Terms & Definitions
- Rational Expression: A fraction with polynomials in the numerator and denominator.
- Limit: The value a function approaches as the input approaches a specific value.
- Continuity: A function is continuous at a point if the function and its limit there agree.
- Derivative: The instantaneous rate of change; slope of the tangent line.
- Chain Rule: Method for differentiating composite functions.
- Implicit Differentiation: Differentiating equations not solved for y.
- Related Rates: Problems involving rates of change of related quantities.
- Integral: Represents accumulated area under a curve or the antiderivative.
- Fundamental Theorem of Calculus: Connects differentiation and integration.
- Critical Number: Value where f'(x) = 0 or is undefined.
- Inflection Point: Where the function changes concavity (f'' changes sign).
- Mean Value Theorem: Guarantees a value where the instantaneous rate matches the average rate on an interval.
- Antiderivative: A function whose derivative is the given function.
- u-Substitution: A method for integrating composite functions by substituting u = g(x).
- Riemann Sum: An approximation of the area under a curve using sums of rectangle areas.
- Differential: An expression representing an infinitesimal change in a function, used for approximations and error estimates.
Action Items / Next Steps
- Practice simplifying, multiplying, dividing, and adding/subtracting rational expressions, including factoring and finding LCDs.
- Review and memorize key limit laws, derivative rules (power, product, quotient, chain), and integral rules, including substitution and integration by parts.
- Solve example problems using the chain rule, implicit differentiation, logarithmic differentiation, and related rates.
- Work through exercises on finding maxima and minima using first and second derivative tests, and identifying inflection points.
- Complete assigned readings and exercises on limits, derivatives, integrals, and the fundamental theorem of calculus, including Riemann sums and average value problems.
- Apply the mean value theorem and the fundamental theorem of calculus to real-world and theoretical problems, including verifying hypotheses and interpreting results.
- Practice using u-substitution and other integration techniques to solve definite and indefinite integrals, and use differentials for error estimation and linear approximations.
- Explore additional applications such as Newton’s method for finding roots, and summation notation for expressing sums and approximating integrals.