Here’s a set of sample notes on Calculus, focusing on key concepts such as limits, derivatives, integrals,

and applications:

1. Limits

 Definition: A limit is the value that a function approaches as the input (or variable) approaches a
certain value.

lim⁡x→af(x)=L\lim_{x \to a} f(x) = L

means that as xx approaches aa, the function f(x)f(x) approaches LL.

 Types of Limits:

o One-sided Limits:

 Left-hand limit: lim⁡x→a−f(x)\lim_{x \to a^-} f(x)

 Right-hand limit: lim⁡x→a+f(x)\lim_{x \to a^+} f(x)

o Infinite Limits: When the function grows without bound as xx approaches a particular
point.

o Limits at Infinity: When x→∞x \to \infty or x→−∞x \to -\infty, the function approaches
a certain value or behaves in a certain way.

 L'Hôpital's Rule: Used for indeterminate forms like 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}.

lim⁡x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

If the limit on the right exists.

2. Derivatives

 Definition: The derivative of a function f(x)f(x) measures the rate of change of f(x)f(x) with
respect to xx. It is the slope of the tangent line to the curve at any point.

f′(x)=lim⁡Δx→0f(x+Δx)−f(x)Δxf'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}

o The derivative represents the instantaneous rate of change of the function.

 Basic Rules for Differentiation:

o Power Rule: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = n x^{n-1}

o Constant Rule: ddx(c)=0\frac{d}{dx}(c) = 0, where cc is a constant.

o Sum and Difference Rule: ddx(f(x)±g(x))=f′(x)±g′(x)\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm

g'(x)

o Product Rule: ddx(f(x)g(x))=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)

 o Quotient Rule: ddx(f(x)g(x))=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx} \left( \frac{f(x)}{g(x)} \
right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

o Chain Rule: For composite functions, ddx(f(g(x)))=f′(g(x))⋅g′(x)\frac{d}{dx}(f(g(x))) =

f'(g(x)) \cdot g'(x)

 Applications of Derivatives:

o Velocity and Acceleration: The derivative of position x(t)x(t) with respect to time tt gives
the velocity v(t)v(t), and the derivative of velocity gives acceleration a(t)a(t).

o Tangent Line: The equation of the tangent line at a point (a,f(a))(a, f(a)) is given by:
y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)

3. Integrals

 Definition: The integral of a function f(x)f(x) represents the area under the curve of f(x)f(x) from
aa to bb.

∫abf(x) dx\int_a^b f(x) \, dx

is the definite integral, and it gives a number representing the total accumulated area between the curve
and the x-axis.

 Indefinite Integral: Represents a family of functions and includes an arbitrary constant CC.

∫f(x) dx=F(x)+C\int f(x) \, dx = F(x) + C

 Basic Integration Rules:

o Power Rule: ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C for n≠−1n \neq -1

o Constant Rule: ∫c dx=cx+C\int c \, dx = cx + C

o Sum Rule: ∫(f(x)±g(x)) dx=∫f(x) dx±∫g(x) dx\int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int
g(x) \, dx

o Substitution Rule: For ∫f(g(x))g′(x) dx\int f(g(x)) g'(x) \, dx, let u=g(x)u = g(x), then du=g′
(x) dxdu = g'(x) \, dx.

 Definite Integral: The value of the integral with limits of integration aa and bb.

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

This represents the signed area between the curve and the x-axis over the interval [a,b][a, b].

 Applications of Integrals:

o Area under a curve: The integral can represent the area between a function and the x-
axis.

o Total Distance: The integral of velocity gives the total distance traveled.
 o Volume: The volume of a solid can be found using integrals, especially in techniques like
the disk or shell method.

4. Fundamental Theorem of Calculus

 First Part: If ff is continuous on [a,b][a, b], then the function F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt
is continuous, differentiable, and F′(x)=f(x)F'(x) = f(x).

 Second Part: If ff is continuous on [a,b][a, b], then:

∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

where F(x)F(x) is any antiderivative of f(x)f(x).

5. Techniques of Integration

 Integration by Parts:

∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Used when the integrand is a product of two functions.

 Partial Fractions: Used to decompose a rational function into simpler fractions that are easier to
integrate.

 Trigonometric Integrals: Involves integrating powers of trigonometric functions, often requiring

specific identities to simplify the process.

 Improper Integrals: These involve limits where the bounds of integration are infinite or the
integrand has a discontinuity within the interval.

6. Applications of Calculus

 Optimization: Finding maximum or minimum values of functions, often using the first derivative
test (set f′(x)=0f'(x) = 0 to find critical points) and second derivative test (to determine concavity
and nature of critical points).

 Related Rates: Solving problems where two or more variables are related and change with
respect to time. For example, finding the rate of change of the area of a circle as its radius
changes.

 Area Between Curves: The area between two curves y=f(x)y = f(x) and y=g(x)y = g(x) over a range
[a,b][a, b] is given by:

∫ab(f(x)−g(x)) dx\int_a^b (f(x) - g(x)) \, dx

 Volume of Solids of Revolution: Using integration to find the volume of solids formed by
rotating a region around an axis. The disk and shell methods are commonly used.
7. Multivariable Calculus (Brief Overview)

 Partial Derivatives: The derivative of a function with more than one variable, holding all but one
variable constant.

o ∂f∂x\frac{\partial f}{\partial x} is the partial derivative of f(x,y,z)f(x, y, z) with respect to

xx.

 Multiple Integrals: Integrals extended to functions of two or more variables.

o Double Integrals: ∫∫Rf(x,y) dx dy\int \int_R f(x, y) \, dx \, dy, used for finding areas and
volumes in two dimensions.

These are some of the fundamental concepts in calculus, and mastering them can provide a solid
foundation for more advanced topics like differential equations, series, and multivariable calculus.