Calculus

The One Thing

Calculus links local change and accumulated change: limits establish how functions behave near points, derivatives measure instantaneous change, and integrals measure accumulation. Differential equations extend these ideas to models of growth, decay, motion, and finance, provided that mathematical results are interpreted with appropriate domains, units, assumptions, and practical restrictions.

Definitions and Results

• Limit: The value that a function approaches as its input approaches a particular number.
• Continuity: A function is continuous at a point if its limit, function value, and left- and right-hand behavior agree there.
• Derivative: The instantaneous rate of change of a function; geometrically, it is the slope of the tangent to the curve.
• Differentiability: A function is differentiable at a point if its derivative exists there. Differentiability implies continuity at that point.
• Higher-Order Derivative: A derivative taken more than once, such as the second derivative, which often describes acceleration or curvature.
• Increasing and Decreasing Functions: A function is increasing where its derivative is positive and decreasing where its derivative is negative. Thus, $f^{\prime }(x)>0$ implies that $f$ is increasing on the interval, while $f^{\prime }(x)<0$ implies that $f$ is decreasing.
• Critical Point: A point in the domain where the derivative is zero or does not exist, and which may indicate a local maximum or minimum.
• Local Maximum and Local Minimum: A local maximum is a nearby highest value, while a local minimum is a nearby lowest value.
• Absolute Maximum and Minimum: The greatest or least value of a function over its entire specified domain.
• Optimization: The process of finding the greatest or least possible value of a quantity subject to given conditions.
• Antiderivative: A function whose derivative is the given function.
• Indefinite Integral: The family of all antiderivatives of a function, written with an arbitrary constant of integration.
• Definite Integral: A signed accumulation of a function over an interval, commonly used to calculate net area or total change.
• Fundamental Theorem of Calculus: Differentiating an accumulated integral returns the original function, and definite integrals can be evaluated using antiderivatives.
• Differential Equation: An equation involving a dependent variable and one or more of its derivatives.
• General Solution: A solution of a differential equation containing arbitrary constants.
• Particular Solution: A solution obtained from the general solution by using given initial or boundary conditions.
• Separable Differential Equation: A differential equation that can be rearranged so that variables and differentials are placed on separate sides.
• Derivative from first principles:

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h},$$ when the limit exists.

• Basic derivative rules:

$$\frac{d}{dx}(c)=0,\frac{d}{dx}(x^n)=n{x}^{n-1},$$ $$\frac{d}{dx}[cf(x)]=c{f}^{\prime }(x),\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x).$$

• Product rule:

$$\frac{d}{dx}[f(x)g(x)]=f(x)g^{\prime }(x)+g(x)f^{\prime }(x).$$

• Quotient rule:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2},$$ where $g(x)\ne 0$.

• Chain rule:

$$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x).$$

• Important derivatives:

$$\frac{d}{dx}(\sin x)=\cos x,\frac{d}{dx}(\cos x)=-\sin x,\frac{d}{dx}(e^x)=e^x,\frac{d}{dx}(\ln x)=\frac{1}{x}(x>0).$$

• Inverse trigonometric derivatives:

$$\frac{d}{dx}({\sin }^{-1}x)=\frac{1}{\sqrt{1-x^2}},\frac{d}{dx}({\cos }^{-1}x)=-\frac{1}{\sqrt{1-x^2}},$$ $$\frac{d}{dx}({\tan }^{-1}x)=\frac{1}{1+x^2},$$ where the expressions are defined.

• Second-derivative test: If $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)>0$, then $f$ has a local minimum at $a$. If $f^{\prime \prime }(a)<0$, then $f$ has a local maximum at $a$.
• Indefinite integral rule:

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1.$$

• Basic integrals:

$$\int \frac{1}{x}dx=\ln |x|+C,\int e^{x}dx=e^x+C,$$ $$\int \sin xdx=-\cos x+C,\int \cos xdx=\sin x+C.$$

• Definite integral evaluation:

$${\int }_a^{b}f(x)dx=F(b)-F(a),$$ where $F^{\prime }(x)=f(x)$.

• Area under a nonnegative curve: For $y=f(x)$ from $x=a$ to $x=b$, the area is

$${\int }_a^{b}f(x)dx.$$ For area between curves, integrate the upper function minus the lower function.

• Substitution method: If $u=g(x)$, then

$$\int f(g(x))g^{\prime }(x)dx=\int f(u)du.$$

• Integration by parts:

$$\int udv=uv-\int vdu.$$

• Average value: The average value of $f$ on $[a,b]$ is

$$\frac{1}{b-a}{\int }_a^{b}f(x)dx.$$

• Separable differential equation: For

$$\frac{dy}{dx}=f(x)g(y),$$ the variables can be separated as $$\frac{dy}{g(y)}=f(x)dx.$$

• Initial condition: An initial condition such as $y(x_0)=y_0$ determines the constant of integration and selects a particular solution.
• Application requirements: A final answer must be interpreted using the relevant domain, units, assumptions, and meaning in the original context.

Worked Methods

Differentiation from first principles

• Begin with the definition

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}.$$

• Substitute the expression for $f(x+h)$ and $f(x)$.
• Simplify the quotient.
• Evaluate the limit as $h\to 0$, provided the limit exists.

This method establishes the derivative as an instantaneous rate of change and as the slope of the tangent to the curve.

Differentiation using derivative rules

• Identify whether the expression contains constants, powers, sums or differences, products, quotients, or composite functions.
• Apply the relevant basic rule:

$$\frac{d}{dx}(c)=0,\frac{d}{dx}(x^n)=n{x}^{n-1},$$ together with the constant multiple and sum or difference rules.

• Use the product rule for $f(x)g(x)$:

$$f(x)g^{\prime }(x)+g(x)f^{\prime }(x).$$

• Use the quotient rule for $f(x)/g(x)$, ensuring that $g(x)\ne 0$.
• Use the chain rule for a composite function $f(g(x))$:

$$f^{\prime }(g(x))g^{\prime }(x).$$

• Apply the listed trigonometric, exponential, logarithmic, and inverse trigonometric derivatives where appropriate.

Finding intervals of increase and decrease

• Find $f^{\prime }(x)$.
• Locate points where $f^{\prime }(x)=0$ or where $f^{\prime }(x)$ does not exist.
• Use these points to divide the domain into intervals.
• Determine the sign of $f^{\prime }(x)$ on each interval.
• State that $f$ is increasing where $f^{\prime }(x)>0$ and decreasing where $f^{\prime }(x)<0$.

Classifying local extrema

• Find the critical points from $f^{\prime }(x)=0$ or from points where $f^{\prime }(x)$ does not exist.
• For a critical point $a$, evaluate $f^{\prime \prime }(a)$, when the second derivative is available.
• If $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)>0$, classify $a$ as a local minimum.
• If $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)<0$, classify $a$ as a local maximum.
• If the second-derivative test does not decide the result, use sign analysis of the first derivative.

Finding absolute extrema on a closed interval

• Determine the critical points in the specified interval.
• Retain only critical points that lie in the domain.
• Evaluate the function at every relevant critical point.
• Evaluate the function at both endpoints.
• Compare all function values.
• The greatest value is the absolute maximum and the least value is the absolute minimum.

Solving an optimization problem

• Translate the practical situation into a mathematical objective function.
• State the constraints and use them to reduce the number of variables.
• Determine the feasible domain of the objective function.
• Differentiate the resulting function.
• Find critical points where the derivative is zero or does not exist.
• Check endpoints and any other boundary values.
• Compare the relevant values to identify the greatest or least quantity.
• Interpret the result in context, including units, assumptions, and practical restrictions.

Evaluating an indefinite integral

• Identify an antiderivative of the integrand.
• Apply the appropriate integration rule, such as

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1.$$

• Include the arbitrary constant $C$.
• Verify the result by differentiating the antiderivative.

The basic cases include $$\int \frac{1}{x}dx=\ln |x|+C,\int e^{x}dx=e^x+C,$$ $$\int \sin xdx=-\cos x+C,\int \cos xdx=\sin x+C.$$

Evaluating a definite integral

• Find an antiderivative $F(x)$ such that $F^{\prime }(x)=f(x)$.
• Apply the Fundamental Theorem of Calculus:

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

• Interpret the result as signed accumulation or net area.
• If the function is below the horizontal axis, account for the resulting negative contribution.

For a nonnegative curve $y=f(x)$, the area from $x=a$ to $x=b$ is $${\int }_a^{b}f(x)dx.$$ For area between curves, subtract the lower function from the upper function before integrating.

Using substitution

• Identify an inner expression $g(x)$ and its derivative.
• Set $u=g(x)$.
• Replace $g(x)$ and $g^{\prime }(x)dx$ with $u$ and $du$.
• Integrate with respect to $u$:

$$\int f(g(x))g^{\prime }(x)dx=\int f(u)du.$$

• Substitute back in terms of $x$, unless the integral is definite and the limits have been changed consistently.

This method reverses the chain rule.

Using integration by parts

• Choose $u$ and $dv$ so that the new integral $\int vdu$ is simpler.
• Find $du$ by differentiating $u$.
• Find $v$ by integrating $dv$.
• Apply

$$\int udv=uv-\int vdu.$$

• Complete and simplify the remaining integral.

Finding an average value

• Identify the interval $[a,b]$.
• Find the definite integral ${\int }_a^{b}f(x)dx$.
• Divide by the interval length:

$$\text{average value}=\frac{1}{b-a}{\int }_a^{b}f(x)dx.$$

Solving a separable differential equation

• Start with

$$\frac{dy}{dx}=f(x)g(y).$$

• Rearrange the variables:

$$\frac{dy}{g(y)}=f(x)dx.$$

• Integrate both sides.
• Include the constant of integration.
• Rearrange to obtain the general solution, if possible.
• Apply an initial or boundary condition such as $y(x_0)=y_0$.
• Determine the constant and state the resulting particular solution.
• Interpret the solution using the domain, units, assumptions, and practical meaning of the model.

Where It Goes Wrong

• Treating a derivative as an average rate rather than an instantaneous rate, or failing to recognise its geometric meaning as the slope of the tangent.
• Applying the quotient rule without retaining the order $g(x)f^{\prime }(x)-f(x)g^{\prime }(x)$, or forgetting the condition $g(x)\ne 0$.
• Using the second-derivative test without first confirming that $f^{\prime }(a)=0$; the test requires $f^{\prime \prime }(a)>0$ for a local minimum or $f^{\prime \prime }(a)<0$ for a local maximum.
• Finding only critical points when determining absolute extrema, while forgetting to check the endpoints of a closed interval.
• Treating a definite integral as ordinary area without accounting for signed accumulation and areas below the horizontal axis.
• Omitting the constant of integration in an indefinite integral, or failing to use an initial condition to determine it in a differential equation.

What Gets Asked

This material supports questions requiring students to:

• Define limits, continuity, derivatives, differentiability, higher-order derivatives, critical points, extrema, antiderivatives, definite integrals, and differential equations.
• Compute derivatives from first principles and by applying basic rules, the product rule, quotient rule, chain rule, and standard trigonometric, exponential, logarithmic, and inverse trigonometric derivatives.
• Determine intervals on which a function is increasing or decreasing using first-derivative sign analysis.
• Classify local maxima and minima using the second-derivative test.
• Find absolute maxima and minima on closed intervals by checking critical points and endpoints.
• Formulate and solve optimization problems subject to constraints.
• Evaluate indefinite and definite integrals, including integrals involving $\frac{1}{x}$, $e^x$, $\sin x$, and $\cos x$.
• Find areas under curves, areas between curves, net accumulation, and average values.
• Evaluate integrals using substitution and integration by parts.
• Solve separable differential equations, distinguish general and particular solutions, and use initial conditions such as $y(x_0)=y_0$.
• Interpret calculus results in practical contexts such as growth, decay, motion, and finance, while stating domains, units, assumptions, and the meaning of the final answer.