Calculus Essentials

you through the foundational ideas of calculus, starting with limits and continuity before moving into the definition of the derivative. The questions range from intuitive statements of what a limit is to the more formal epsilon-delta version, and they cover key theorems like the Squeeze Theorem and the Intermediate Value Theorem. You'll also practice recognizing common situations such as removable discontinuities, one-sided limits, and the famous limit of sin(x)/x as x approaches zero.

It's a great fit for students in an early calculus course who want to firm up the conceptual groundwork before diving into computational techniques, or for anyone returning to the subject who wants to refresh the theory behind derivatives. Because many of the cards focus on precise definitions, this deck is also useful for test prep where being able to state theorems and definitions word-for-word matters.

Because the content is definition-heavy, try to answer each card from memory before flipping — paraphrasing what "continuity at a point c" means in your own words is just as valuable as recalling the exact phrasing. Spacing your review over a few short sessions rather than cramming everything at once will help these formal statements stick, especially the more technical ones like the epsilon-delta definition of a limit.

The limit is the foundational concept that underlies all of calculus. Informally, the limit of \(f(x)\) as \(x\) approaches \(c\) is the value \(L\) that \(f(x)\) approaches as \(x\) gets arbitrarily close to \(c\). More precisely, we write \(\lim_{x \to c} f(x) = L\) to mean that \(f(x)\) can be made as close to \(L\) as we like by choosing \(x\) sufficiently near \(c\). This intuitive idea is captured rigorously by the epsilon-delta definition: for every \(\varepsilon > 0\), there must exist a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \varepsilon\). The rigor of this formulation is what allows calculus to be placed on solid logical footing.

A function is continuous at a point \(c\) precisely when three conditions hold: \(f(c)\) is defined, \(\lim_{x \to c} f(x)\) exists, and that limit equals \(f(c)\). Graphically, a continuous function is one whose graph can be drawn through \(c\) without lifting the pen. Discontinuities arise when one or more of these conditions fails; a particularly interesting case is the removable discontinuity, which occurs when the limit exists at \(c\) but either \(f(c)\) is undefined or \(f(c) \neq \lim_{x \to c} f(x)\). Such a discontinuity can be removed by redefining \(f(c)\) to equal the limit value. Closely related are one-sided limits, written \(\lim_{x \to c^+} f(x)\) for the right-hand limit and \(\lim_{x \to c^-} f(x)\) for the left-hand limit; the two-sided limit exists only when both one-sided limits agree.

Several important theorems build on the notion of limits. The Squeeze Theorem states that if \(g(x) \leq f(x) \leq h(x)\) near \(c\) and both \(g\) and \(h\) share the same limit \(L\) at \(c\), then \(f\) must also have limit \(L\). This is how one proves the fundamental limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), using geometric comparisons with the unit circle. A practical matter is the interpretation of infinite limits: writing \(\lim_{x \to c} f(x) = \infty\) does not mean the limit equals infinity in the usual sense; rather, \(f(x)\) grows without bound as \(x\) approaches \(c\), exceeding every positive number eventually.

When limits are taken as \(x\) grows arbitrarily large, we examine end behavior. For rational functions \(P(x)/Q(x)\) as \(x \to \infty\), the standard technique is to divide every term by the highest power of \(x\) appearing in the denominator. After this division, every term containing \(x\) in the denominator approaches 0, leaving only the ratio of leading coefficients when the numerator and denominator have the same degree. The Intermediate Value Theorem rounds out this chapter: if \(f\) is continuous on a closed interval \([a, b]\), then \(f\) attains every value between \(f(a)\) and \(f(b)\) somewhere in \((a, b)\). This theorem is foundational for many existence arguments in calculus and beyond.

The derivative measures instantaneous rate of change. Formally, the derivative of \(f\) at a point \(x\) is defined as \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, \] provided this limit exists. An equivalent formulation focuses on a specific point \(a\): \[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}. \] These two forms are interchangeable and can be applied based on whichever is more convenient. The derivative exists at a point only when this limit is finite and well-defined; if the function has a corner, cusp, vertical tangent, or discontinuity at a point, the limit fails to exist and the derivative is undefined there.

Geometrically, the derivative \(f'(a)\) gives the slope of the tangent line to the graph of \(y = f(x)\) at the point \((a, f(a))\). The tangent line is the best linear approximation to the function near \(a\), meaning it matches the function's instantaneous direction at that point. Physically, if \(s(t)\) denotes the position of an object at time \(t\), then \(s'(t)\) represents the instantaneous velocity, and \(s''(t)\) the instantaneous acceleration. More broadly, the derivative captures the rate at which one quantity changes with respect to another.

An important logical relationship connects differentiability and continuity: if \(f\) is differentiable at a point, then \(f\) is necessarily continuous there as well. However, the converse fails. The classic counterexample is \(f(x) = |x|\) at \(x = 0\), where the function is continuous (since \(|0| = 0\) and the limit equals 0), but the sharp corner prevents the derivative from existing. This asymmetry between continuity and differentiability is one of the subtle but important ideas at the heart of calculus.

Computing derivatives directly from the limit definition is tedious, so a collection of rules streamlines the process. The Power Rule states that if \(f(x) = x^n\) for any real exponent \(n\), then \(f'(x) = n x^{n-1}\). This rule applies to integer, fractional, and negative exponents alike, making it one of the most frequently used results in differentiation. For products of functions, the Product Rule applies: if \(f(x) = u(x) \cdot v(x)\), then \(f'(x) = u'(x) v(x) + u(x) v'(x)\). The Quotient Rule handles ratios: if \(f(x) = u(x)/v(x)\), then \[ f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}. \]

The Chain Rule governs compositions of functions. If \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\), or equivalently in Leibniz notation, \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). The Chain Rule is essential whenever one function is applied inside another, and it underlies many derivative formulas. For example, to differentiate \(f(x) = a^x\) for a positive constant \(a\), one rewrites \(a^x = e^{x \ln a}\) and applies the Chain Rule, yielding \(f'(x) = a^x \ln a\). When \(a = e\), this gives the elegant result \(f'(x) = e^x\), showing that the exponential function is its own derivative.

Trigonometric derivatives are immediate consequences of the basic rules. The derivative of \(\sin x\) is \(\cos x\), derivable from the limit definition together with the fundamental limits \(\lim_{h \to 0} \frac{\sin h}{h} = 1\) and \(\lim_{h \to 0} \frac{\cos h - 1}{h} = 0\). Differentiating \(\cos x\) gives \(-\sin x\), while \(\tan x = \frac{\sin x}{\cos x}\) has derivative \(\sec^2 x\) by the Quotient Rule. For the natural logarithm, \(\frac{d}{dx} \ln x = \frac{1}{x}\) for \(x > 0\), a result that follows from the fact that \(\ln x\) is the inverse of \(e^x\) and can be derived using implicit differentiation. The derivative of \(\arcsin x\) is \(\frac{1}{\sqrt{1 - x^2}}\) for \(|x| < 1\), obtained through implicit differentiation of the relation \(\sin y = x\).

Sometimes the relationship between \(x\) and \(y\) is given implicitly by an equation like \(x^2 + y^2 = 25\) rather than explicitly as \(y = f(x)\). In such cases, implicit differentiation provides a way to find \(dy/dx\). One differentiates both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\) and applying the Chain Rule whenever \(y\) appears. For example, from \(x^2 + y^2 = 25\) we get \(2x + 2y \frac{dy}{dx} = 0\), which simplifies to \(\frac{dy}{dx} = -\frac{x}{y}\). This technique is especially valuable when the equation cannot be solved explicitly for \(y\), as with curves like \(x^3 + y^3 = 6xy\), and when dealing with inverse trigonometric functions. Logarithmic differentiation is a related technique useful for complicated expressions: one takes the natural log of both sides, differentiates implicitly, and then solves for \(dy/dx\). Functions like \(y = x^x\) and products or quotients of many factors are best handled this way.

The derivative is the foundation for optimization and extremum analysis. A critical point of \(f\) occurs at \(x = c\) in the domain of \(f\) whenever \(f'(c) = 0\) or \(f'(c)\) does not exist. Every local maximum or minimum must occur at a critical point, making these the natural candidates to check. The First Derivative Test classifies a critical point: if \(f'\) changes from positive to negative at \(c\), then \(f\) has a local maximum there; if \(f'\) changes from negative to positive, then \(f\) has a local minimum; if no sign change occurs, the point is neither. The Second Derivative Test offers an alternative: if \(f'(c) = 0\) and \(f''(c) > 0\), then \(c\) is a local minimum; if \(f''(c) < 0\), then \(c\) is a local maximum. When \(f''(c) = 0\), the test is inconclusive.

The general strategy for optimization problems is to identify the quantity to optimize, express it as a function of a single variable using the given constraints, find critical points by setting the derivative to zero, and verify the result using one of the derivative tests. Related rates problems follow a similar structure: given an equation relating several time-dependent quantities, differentiate implicitly with respect to time, substitute the known rates, and solve for the unknown. For instance, if a spherical balloon's volume grows at 10 cm³/s, the formula \(V = \frac{4}{3}\pi r^3\) implies \(\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\), so when \(r = 5\) cm, the radius increases at \(\frac{1}{10\pi} \approx 0.0318\) cm/s. In a sliding ladder problem, the Pythagorean relation \(x^2 + y^2 = 100\) yields the speed at which the top descends once the rates of one variable and the values of both are known.

Two deeper results round out the applications of derivatives. The Mean Value Theorem states that if \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there is at least one point \(c \in (a, b)\) where \(f'(c) = \frac{f(b) - f(a)}{b - a}\). Geometrically, this says that somewhere on the graph the tangent line is parallel to the secant line joining the endpoints. Concavity describes how the graph bends: \(f\) is concave up on an interval where \(f''(x) > 0\), with the graph curving upward like a cup, and concave down where \(f''(x) < 0\), with the graph curving downward like a cap. Inflection points mark where the concavity changes from one to the other, and these often coincide with points where \(f''(x) = 0\) or fails to exist.

Integration can be viewed as the inverse of differentiation. An indefinite integral \(\int f(x) \, dx\) represents the family of all antiderivatives of \(f\), expressed as \(F(x) + C\), where \(F'(x) = f(x)\) and \(C\) is the constant of integration capturing the vertical shifts in this family. A definite integral \(\int_a^b f(x) \, dx\) has a more concrete meaning: it represents the signed area between the graph of \(f\) and the \(x\)-axis from \(x = a\) to \(x = b\), with regions above the \(x\)-axis contributing positive area and those below contributing negative area. Together, these two notions of integration are connected by the Fundamental Theorem of Calculus.

The Power Rule for integration reverses the corresponding differentiation rule: \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for any \(n \neq -1\). When \(n = -1\), the special case \(\int \frac{1}{x} \, dx = \ln |x| + C\) applies, where the absolute value is needed because \(1/x\) is defined for negative inputs even though \(\ln x\) is not. Other essential antiderivatives include \(\int e^x \, dx = e^x + C\) (the exponential function is its own antiderivative, just as it is its own derivative), \(\int \sin x \, dx = -\cos x + C\), \(\int \cos x \, dx = \sin x + C\), and \(\int \sec^2 x \, dx = \tan x + C\).

The First Fundamental Theorem of Calculus states that if \(f\) is continuous on \([a, b]\) and \(F\) is any antiderivative of \(f\), then \[ \int_a^b f(x) \, dx = F(b) - F(a). \] This allows definite integrals to be evaluated without resorting to limit-of-Riemann-sums calculations. The Second Fundamental Theorem goes further: if \(f\) is continuous and \(F(x) = \int_a^x f(t) \, dt\), then \(F'(x) = f(x)\). In other words, differentiating an integral with a variable upper limit recovers the original integrand. So \(F(x) = \int_0^x \cos(t^2) \, dt\) has derivative \(\cos(x^2)\), even though no elementary antiderivative of \(\cos(t^2)\) is known.

When the upper limit itself is a function of \(x\), the Chain Rule combines with the Second Fundamental Theorem to give \(\frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)\). This composite result is essential in many advanced applications. The Fundamental Theorem is regarded as one of the crowning achievements of calculus because it unifies differentiation and integration as inverse processes and provides a practical method for computing definite integrals without Riemann sums, while also establishing that every continuous function has an antiderivative.

For many integrals, no single antiderivative rule applies directly, and one must transform the integral into a more tractable form. The most common such technique is u-substitution, which reverses the Chain Rule. One identifies a portion of the integrand as \(u = g(x)\), computes the differential \(du = g'(x) \, dx\), and rewrites the integral as \(\int f(u) \, du\). After integrating in \(u\), the result is expressed in terms of \(x\). For example, \(\int 2x \cos(x^2) \, dx\) becomes \(\int \cos(u) \, du\) with \(u = x^2\), giving \(\sin(x^2) + C\). For definite integrals, the limits of integration must also be converted when performing substitution; if \(u = g(x)\), then the original limits \(x = a\) and \(x = b\) become \(u = g(a)\) and \(u = g(b)\), eliminating the need to substitute back at the end.

When the integrand is a product and direct integration is difficult, integration by parts is the technique of choice. Derived from the product rule for differentiation, it states that \(\int u \, dv = uv - \int v \, du\). The goal is to choose \(u\) and \(dv\) so that the new integral \(\int v \, du\) is simpler than the original. A useful guideline for selecting \(u\) is the LIATE rule, prioritizing in order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. The function higher on the list is typically chosen as \(u\). For instance, \(\int x e^x \, dx\) is best handled by setting \(u = x\) and \(dv = e^x \, dx\), giving \(\int x e^x \, dx = xe^x - e^x + C = e^x(x - 1) + C\). Similarly, \(\int \ln x \, dx = x \ln x - x + C\).

Sometimes two applications of parts are needed: integrating \(x^2 e^x\) requires parts twice, while integrals like \(\int e^x \sin x \, dx\) produce the original after two applications, allowing one to solve algebraically for the answer. For integrating rational functions \(P(x)/Q(x)\), the method of partial fractions is used. The denominator is factored into linear or irreducible quadratic pieces, and the original fraction is decomposed into a sum of simpler fractions with these as denominators; each piece is then integrated individually. Many trigonometric integrals can be reduced by identities, especially half-angle formulas. For example, \(\int \sin^2 x \, dx\) begins with the identity \(\sin^2 x = \frac{1 - \cos(2x)}{2}\), giving \(\frac{x}{2} - \frac{\sin(2x)}{4} + C\). Combining u-substitution, integration by parts, partial fractions, and trigonometric identities allows a wide variety of integrals to be evaluated.

Integration gives a precise language for measuring accumulation, with practical applications in geometry. The area between two curves \(y = f(x)\) and \(y = g(x)\) is given by \(\int_a^b |f(x) - g(x)| \, dx\), where \(a\) and \(b\) are the points of intersection. When \(f(x) \geq g(x)\) throughout, this simplifies to \(\int_a^b [f(x) - g(x)] \, dx\). The first step in such problems is finding the intersection points by setting \(f(x) = g(x)\) and solving. For example, the area between \(y = x^2\) and \(y = x\) on \([0, 1]\) is \(\int_0^1 (x - x^2) \, dx = \frac{1}{6}\). If the curves cross within the interval, the integral must be split at each crossing point so that positive and negative contributions do not cancel. When the curves are easier to describe as functions of \(y\), one can integrate with respect to \(y\) instead, using \(\int_c^d |f(y) - g(y)| \, dy\) with horizontal slices.

When a region is revolved around an axis, the resulting three-dimensional solid has a volume expressible as an integral. The disk method applies when cross sections perpendicular to the axis of rotation are solid disks: \(V = \pi \int_a^b [R(x)]^2 \, dx\), where \(R(x)\) is the distance from the curve to the axis. The washer method handles solids with a hole, giving \(V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) \, dx\) with outer radius \(R\) and inner radius \(r\). When the axis of rotation is perpendicular to the variable of integration, the shell method is often easier: \(V = 2\pi \int_a^b x \cdot f(x) \, dx\) for revolution about the \(y\)-axis, where each cylindrical shell has radius \(x\), height \(f(x)\), and thickness \(dx\). For example, revolving \(y = \sqrt{x}\) from \(x = 0\) to \(x = 4\) about the \(x\)-axis gives \(V = \pi \int_0^4 x \, dx = 8\pi\). Revolving about a non-standard axis simply adjusts the radius function to measure distance from the new axis.

Indeterminate forms arise when limits cannot be evaluated by direct substitution. L'Hopital's Rule addresses the cases \(0/0\) and \(\infty/\infty\): if \(\lim_{x \to c} f(x)/g(x)\) is indeterminate of this kind, then the limit equals \(\lim_{x \to c} f'(x)/g'(x)\), provided this latter limit exists. The rule can be applied repeatedly if necessary. Other indeterminate forms \((0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(1^\infty\), \(\infty^0)\) must first be rewritten into \(0/0\) or \(\infty/\infty\) form through algebraic manipulation. For instance, \(\lim_{x \to 0^+} x \ln x\) is rewritten as \(\lim_{x \to 0^+} \frac{\ln x}{1/x}\), after which L'Hopital's Rule applies to give 0. The rule also confirms that \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and shows that exponential growth dominates any polynomial: \(\lim_{x \to \infty} \frac{x}{e^x} = 0\).

A series \(\sum_{n=1}^{\infty} a_n\) assigns a value to an infinite sum by means of partial sums. If the partial sums \(S_n = a_1 + a_2 + \cdots + a_n\) approach a finite limit as \(n \to \infty\), the series converges; otherwise, it diverges. Several standard series serve as benchmarks. The geometric series \(\sum_{n=0}^{\infty} a r^n\) converges to \(\frac{a}{1-r}\) whenever \(|r| < 1\) and diverges otherwise. The \(p\)-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges when \(p > 1\) and diverges when \(p \leq 1\); the harmonic series \(\sum \frac{1}{n}\) is the borderline case that diverges. The Divergence Test gives a quick check: if \(\lim_{n \to \infty} a_n \neq 0\), the series must diverge, although the test is inconclusive when the limit is 0.

More powerful tests determine convergence in trickier cases. The Integral Test compares a series \(\sum a_n\) with the corresponding integral \(\int_1^\infty f(x) \, dx\), where \(a_n = f(n)\): if \(f\) is positive, continuous, and decreasing, then the series and the integral either both converge or both diverge. The Comparison Test uses known series for reference: if \(0 \leq a_n \leq b_n\) and \(\sum b_n\) converges, then \(\sum a_n\) converges. The Ratio Test computes \(L = \lim_{n \to \infty} |a_{n+1}/a_n|\); convergence is assured if \(L < 1\) and divergence if \(L > 1\), with \(L = 1\) inconclusive. The Root Test is similar, using \(L = \lim_{n \to \infty} |a_n|^{1/n}\) with the same thresholds. These last two are particularly suited to series involving factorials or \(n\)-th powers.

Alternating series, with terms that switch sign like \(\sum (-1)^n b_n\), are handled by the Alternating Series Test: such a series converges whenever \(b_n\) is decreasing and \(\lim_{n \to \infty} b_n = 0\). This makes possible conditional convergence, where \(\sum a_n\) converges but \(\sum |a_n|\) diverges (the alternating harmonic series \(\sum (-1)^{n+1}/n\) being a classic example, equal to \(\ln 2\)). Absolute convergence, where \(\sum |a_n|\) converges, is the stronger condition that always implies convergence. Distinguishing these two notions is essential in advanced analysis.

Taylor and Maclaurin series extend calculus to represent functions as infinite polynomials. The Taylor series of \(f\) centered at \(x = a\) is \[ \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n, \] and a Maclaurin series is simply the special case \(a = 0\). Among the most important are \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\), \(\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\), and \(\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\), all convergent for every real \(x\). The radius of convergence \(R\) of a power series \(\sum a_n (x - c)^n\) is the threshold such that the series converges for \(|x - c| < R\) and diverges for \(|x - c| > R\); it can be found using the Ratio Test. These series provide a powerful bridge between functions and polynomials, enabling approximation and computation in settings where direct evaluation is impractical.