Comprehensive Guide to Differential Calculus

A set of practice flashcards covering key concepts, rules, and theorems from the differential calculus notes (limits, continuity, derivatives, and applications).

What is a limit?

Describes the value a function approaches as x approaches a specific point.

What does continuity mean at a point?

The function is defined at that point, the limit as x approaches the point exists, and the limit equals the function value.

What are the three conditions for f to be continuous at x = c?

1) f(c) is defined; 2) lim x→c f(x) exists; 3) lim x→c f(x) = f(c).

What are the one-sided limits?

lim x→c− f(x) (from the left) and lim x→c+ f(x) (from the right).

Name the types of discontinuities.

Jump discontinuity, Infinite discontinuity, and Removable discontinuity.

How can you evaluate limits analytically?

Using substitution, factoring, and rationalization to simplify expressions.

What are common indeterminate forms?

0/0 and ∞/∞.

How is the derivative defined?

f′(a) = lim h→0 [f(a+h) − f(a)]/h, the slope of the tangent line at x = a.

What does the derivative tell you?

The instantaneous rate of change and signs of increasing/decreasing behavior, helping identify local extrema.

State the Power Rule.

d/dx x^n = n x^(n−1).

State the Constant Rule.

d/dx c = 0, where c is a constant.

State the Constant Multiplier Rule.

d/dx [k · f(x)] = k · f′(x) for constant k.

What is implicit differentiation?

Differentiate both sides with respect to x when y isn’t isolated; use the chain rule for dy/dx.

What are related rates?

Problems involving quantities that change over time; differentiate with respect to time and use the chain rule to relate variables.

What are absolute extrema?

The highest or lowest values of a function on a closed interval.

What are local extrema?

Local maximum or minimum values within a neighborhood of a point.

What does the Mean Value Theorem (MVT) say?

If f is continuous on [a,b] and differentiable on (a,b), there exists c ∈ (a,b) with f′(c) = (f(b) − f(a)) / (b − a).

First Derivative Test: when is there a local maximum or minimum?

If f′ changes from positive to negative at c, there is a local maximum; if f′ changes from negative to positive at c, there is a local minimum.

Second Derivative Test: criteria and inconclusive case.

If f′′(c) > 0, local minimum; if f′′(c) < 0, local maximum; if f′′(c) = 0, the test is inconclusive.

What does the second derivative tell you about concavity and inflection points?

If f′′(x) > 0, concave up; if f′′(x) < 0, concave down; inflection points occur where concavity changes (f′′ changes sign).

What are horizontal asymptotes?

A horizontal line y = L that the graph approaches as x → ±∞; if lim x→∞ f(x) = L, then y = L is a horizontal asymptote (similarly for x → −∞).

How do you approach optimization problems with derivatives?

Find critical points by solving f′(x) = 0 (or undefined); use the first or second derivative test to classify; apply to real-world objectives.

Derivatives of logarithmic and exponential functions: key formulas (d/dx ln x).

d/dx ln x = 1/x.

Derivatives of logarithmic and exponential functions: base-a logarithm and exponential.

d/dx log_a x = 1/(x ln a) for a > 0, a ≠ 1; d/dx a^x = a^x ln a.

Derivatives of inverse trigonometric functions.

d/dx arcsin x = 1/√(1 − x^2); d/dx arccos x = −1/√(1 − x^2); d/dx arctan x = 1/(1 + x^2).

L’Hôpital’s Rule for indeterminate forms.

If a limit gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x), provided the latter limit exists.