Calculus 1 Mastered

Explore the definition of a function, learn how to evaluate it, and review domain, range, exponential, and logarithmic functions as foundational topics for limits in calculus i.

Define functions as a machine that maps each input (domain) to exactly one output (range) using a rule like f(x)=x^2, ensuring unique associations.

Identify the range of a function by contrasting it with the domain, treat functions as machines, and determine that f(x) = x^2 produces nonnegative outputs.

Explore natural logarithms, their key properties and domains, including ln(1)=0, ln e=1, ln(e^x)=x, and domain rules for expressions like ln(x+3) and ln(3-x).

Learn how one-sided limits define the behavior of a function as x approaches a, using right-hand and left-hand limits and examples like step functions.

Explore limit techniques for rational functions, including factoring and cancellation, and the conjugate method to handle square roots, to solve diverse limit problems.

Apply the squeeze theorem to evaluate limits. Bound x^2 sin(1/x) by -x^2 and x^2 to get limit 0, and bound x cos(3/x) - 5 to get limit -5.

Students learn the precise definition of limit through delta-epsilon proofs, identifying f(x), l, c, and deriving delta from epsilon, illustrated with three examples.

Evaluate limits at infinity by comparing degrees and leading coefficients, and by dividing by the highest power to identify dominant terms. Apply these rules to rational and square root expressions.

Explore infinite limits and vertical asymptotes by examining how f(x) behaves near a point, demonstrating right and left limits approaching positive or negative infinity.

Explore the definition and conditions for continuity, examine the properties of continuous functions, and study limits and the intermediate value theorem for continuous functions.

Define continuity by ensuring the limit of f(x) as x approaches c exists and equals f(c), confirming the three conditions for a continuous function.

Explore the fundamentals of derivatives by examining rate of change and average rate of change, then learn the definition and rules of derivatives to differentiate functions, with attached practice downloads.

Explore instantaneous rate of change as the limit of the average rate of change, yielding the derivative f'(x1) and the tangent line slope at a point.

Master the constant rule, derivative of a constant is zero, and the power rule, bring the exponent down and subtract one.

Apply the constant multiple rule to differentiate c·f(x) by leaving the constant untouched and using the power rule on f(x), illustrated with 5x^4, 2x, and 5x^-5.

Master the quotient rule for differentiating a function as a quotient, h(x)=f(x)/g(x). Compute h'(x) = (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2, and apply it in examples with f and g.

Learn to sketch the graph of a function's derivative by identifying zeros, sign patterns, and whether the slope is increasing or decreasing, with practice examples.

Learn how to differentiate trig functions, including sine, cosine, tangent, secant, cosecant, and cotangent, with rules and worked examples using product and quotient rules.

We minimize the cylinder's surface area subject to a fixed volume of one liter, deriving r = (500/pi)^(1/3) and h = 2r for the least material cost.

Apply the second derivative test to critical points to identify local extrema. If f''<0, it's a local max; if f''>0, it's a local min; if f''=0, the test fails.

Learn a five-step method to graph functions using domain, symmetries, asymptotes, intercepts, and first/second derivatives, illustrated by f(x)=x^4-2x^2 and its concavity and inflection points.

Master graphing a rational function using five steps: determine domain, asymptotes, intercepts, and symmetry; analyze derivatives for critical points and concavity to sketch the graph.

Master the mean value theorem: continuous on [a,b], differentiable on (a,b), guarantees a c where the average rate of change equals the instantaneous rate; Rolle's theorem is a special case.

Apply the mean value theorem to a 200-kilometer trip, showing that an average speed of 100 km/h implies a moment when speed equals 100 km/h, explaining a speeding ticket.