Last Minute Notes (LMNs) - Calculus and Optimization

In engineering mathematics, calculus is one of the important branches of mathematics from which questions are asked in the GATE exam, including CSE and DA. This article covers all the key topics within calculus that are frequently tested in the exam and provides links to further resources for each topic.

Function of Single Variable

A function f of a single variable x is written as:

f(x) = expression involving x

For each x in the domain of f, there is a unique value f(x) in the range of the function.

• Domain: The set of all possible input values (x) for which the function is defined.
  • Example: For f(x) = 1/x​, the domain is x ≠ 0 (all real numbers except 0).
• Range: The set of all possible output values (f(x)).
  • Example: For f(x) = x², the range is f(x) ≥ 0 (all non-negative real numbers).

Read more about Domain and Range of Function.

Limits

The limits of a function f(x) as x approaches a value c is written as:

\lim_{x \to c} f(x) = L

This means that as x gets arbitrarily close to c, the function f(x) approaches L.

Right-Hand Limit

• As x approaches c from the right (x→c⁺) i.e., \lim_{x \to c^+} f(x).

Left-Hand Limit

• As x approaches c from the left (x→c⁻) i.e., \lim_{x \to c^-} f(x).

The limit exists only if the right-hand and left-hand limits are equal:

\lim_{x \to c} f(x) = \lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)

Formally, we can say that:

\lim_{x \to c} f(x) = L

If for every ϵ > 0 (no matter how small), there exists a δ > 0 such that whenever 0 < ∣x − c∣ < δ, it follows that ∣f(x) − L∣ < ϵ|.

Also, Read Formal Definition of Limit.

Properties of Limits

Some common properties of limits are:

• Addition: \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)
• Multiplication by a Constant: \lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)
• Product: \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)
• Quotient (if the denominator is non-zero): \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c}}
• Power: \lim_{x \to c} [f(x)]^n = \left(\lim_{x \to c} f(x)\right)^n

Read More about the Properties of Limits.

Some Common Limits

Some of the common limits used in calculus are:

\bullet\: \lim_{x\to 0} \frac{\sin x}{x} = 1 \\\bullet\: \lim_{x\to 0} \cos x = 1 \\\bullet\: \lim_{x\to 0} \frac{\tan x}{x} = 1 \\\bullet\: \lim_{x\to 0} \frac{1-\cos x}{x} = 0 \\\bullet\: \lim_{x\to 0} \frac{\sin x^\circ}{x} = \frac{\pi}{180} \\\bullet\: \lim_{x\to a} \frac{x^n - a^n}{x-a} = na^{n-1} \\\bullet\: \lim_{x\to \infty} \left(1+\frac{k}{x}\right)^{mx} = e^{mk} \\\bullet\: \lim_{x\to 0} (1+x)^{\frac{1}{x}} = e \\\bullet\: \lim_{x\to 0} \frac{a^x-1}{x} = \ln a \\\bullet\: \lim_{x\to 0} \frac{e^x-1}{x} = 1 \\\bullet\: \lim_{x\to 0} \frac{\ln (1+x)}{x} = 1 \\\bullet\: \lim_{x\to \infty} x^{\frac{1}{x}} = 1

L'Hospital Rule

If the given limit \lim_{x\to a} \frac{f(x)}{g(x)} is of the form \frac{0}{0} or \frac{\infty}{\infty} i.e. both f(x) and g(x) are either 0 or ∞, then the limit can be solved by L'Hospital Rule. 

If the limit is of the form described above, then the L'Hospital Rule says that:

\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f^\prime(x)}{g^\prime(x)}

Where f'(x) and g'(x) are obtained by differentiating f(x) and g(x). If after differentiating, the form still exists, then the rule can be applied continuously until the form is changed.

Squeeze Theorem

Squeeze Theorem (also called the Sandwich Theorem) works by "squeezing" a function between two others whose limits are known and equal at a particular point.

If g(x) ≤ f(x) ≤ h(x) for all x in some interval around c (except possibly at c itself), and if \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L, then:

\lim_{x \to c} f(x) = L

Continuity

A function f(x) is continuous at a point x = c if the following three conditions are satisfied:

1. The function is defined at c: f(c) exists.
2. The limit of the function exists as x→c: lim_⁡x→cf(x) exists.
3. The value of the function matches the limit: lim⁡_x→cf(x) = f(c).

If any of these conditions fail, the function is not continuous at x = c.

Continuity on an Interval

• A function f(x) is said to be continuous on an interval [a, b] if it is continuous at every point in the interval.
• At the endpoints of a closed interval, [a, b], continuity is defined as:
  • At x = a: \lim_{x \to a^+} f(x) = f(a).
  • At x=bx = bx=b: \lim_{x \to b^-} f(x) = f(b).

Also Read about Continuity at a Point.

Functions that are not continuous are said to be discontinuous.

Types of Discontinuity

If a function is not continuous at a point, it is said to have a discontinuity at that point. There are several types:

• Removable Discontinuity:
  • A "hole" in the graph.
  • The limit exists, but f(c) is either not defined or does not equal lim_⁡x→cf(x).
• Jump Discontinuity:
  • The left-hand limit (\lim_{x \to c^-}​) and the right-hand limit (\lim_{x \to c^+}​) exist but are not equal.
• Infinite Discontinuity:
  • The function approaches infinity or negative infinity as x→c.
• Oscillatory Discontinuity:
  • The function oscillates wildly as x→c, and the limit does not exist.

Differentiability

A function f(x) is differentiable at x = c if the following limit exists:

f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}

Here:

• f′(c): The derivative of f(x) at x = c, which represents the slope of the tangent line to the graph of f(x) at that point.
• h: The small change in x.

Note: f'(c) is the derivative of function f(x) at x = c.

A function is said to be differentiable if the derivative of the function exists at all points of its domain.

Note: If a function is differentiable at a point, then it is also continuous at that point, but if a function is continuous at a point does not imply that the function is also differentiable at that point. For example, f(x) = |x| is continuous at x = 0 but it is not differentiable at that point.

Properties of Differentiation

Some common properties or rules of differentiation are:

• \frac{d}{dx}[k \cdot f(x)] = k \cdot f'(x), where k is constant.
• \frac{d}{dx}[f(x) \pm g(x)] = f'(x) + g'(x)
• \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
• \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
• \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Differentiation Formulas

Some of the most common formula used to find derivative are tabulated below:

Mean Value Theorems

Some mean value theorems are:

• Rolle’s Mean Value Theorem
• Lagrange’s Mean Value Theorem
• Cauchy's Mean Value Theorem

Rolle’s Mean Value Theorem

Suppose f(x) be a function satisfying three conditions:

• f(x) is continuous in the closed interval a ≤ x ≤ b
• f(x) is differentiable in the open interval a < x < b
• f(a) = f(b)

Then according to Rolle's Theorem, there exists at least one point 'c' in the open interval (a, b) such that:

f '(c) = 0

Lagrange’s Mean Value Theorem

Suppose f:[a,b]\rightarrow R be a function satisfying three conditions: 

• f(x) is continuous in the closed interval a ≤ x ≤ b
• f(x) is differentiable in the open interval a < x < b

Then according to Lagrange's Theorem, there exists at least one point 'c' in the open interval (a, b) such that:

f'(c)=\frac{f(b)-f(a)}{b-a}

Cauchy's Mean Value Theorem

Let f(x) and g(x) be two functions that satisfy the following conditions:

1. f(x) and g(x) are continuous on the closed interval [a, b],
2. f(x) and g(x) are differentiable on the open interval (a, b),
3. g′(x) ≠ 0 for all x ∈ (a, b).

Then, there exists at least one c ∈ (a, b) such that:

\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}

Maxima and Minima

• Critical Points: Points where the derivative f′(x) = 0 or f′(x) is undefined.
• Local Maximum: f(x) has a local maximum at x = c if f(c) ≥ f(x) for all x in a small neighborhood around c.
• Local Minimum: f(x) has a local minimum at x = c if f(c) ≤ f(x) for all x in a small neighborhood around c.
• Global Maximum/Minimum: The highest/lowest value of f(x) over its entire domain.

Read More about Maxima and Minima.

First Derivative Test:

• If f′(x) changes:
  • From positive to negative at x = c: Local Maximum at c.
  • From negative to positive at x = c: Local Minimum at c.
  • If no change: x = c is a point of inflection.

Second Derivative Test:

• Compute the second derivative f′′(x):
  • If f′′(c) > 0: Local Minimum at c.
  • If f′′(c) < 0: Local Maximum at c.
  • If f′′(c) = 0: Test is inconclusive (use the first derivative test or higher-order derivatives).

Concavity:

• f(x) is:
  • Concave Up if f′′(x) > 0.
  • Concave Down if f′′(x) < 0.
• Point of inflection: f(x) changes concavity (where f′′(x) = 0.

Maxima and Minima in Multivariable Functions:

For f(x, y):

1. Critical Points: Solve ∂f/∂x = 0 and ∂f/∂y = 0.
2. Second Partial Derivatives: Compute:
   • f_xx = ∂²f/∂x²​,
   • f_yy = ∂²f/∂y²,
   • fxy = ∂²f/∂x∂y.
3. Hessian Determinant: H = f_xxf_yy − (f_xy)².
   • If H > 0 and f_xx > 0: Local Minimum.
   • If H > 0 and f_xx < 0: Local Maximum.
   • If H < 0: Saddle Point.
   • If H = 0: Test is inconclusive.

Integrals

Integrals can be classified as:

• Definite Integral
• Indefinite Integral
• Improper Integrals

Indefinite Integrals

Let f(x) be a function. Then the family of all its antiderivatives is called the indefinite integral of a function f(x) and it is denoted by ∫f(x)dx.

• The symbol ∫f(x)dx is read as the indefinite integral of f(x) with respect to x. Thus ∫f(x)dx= ∅(x) + C. Thus, the process of finding the indefinite integral of a function is called integration of the function.

Fundamental Integration Formulas:

Some common integration formulas include:

1. ∫xⁿdx = (xⁿ⁺¹/(n+1))+C
2. ∫(1/x)dx = (log_e|x|)+C
3. ∫e^xdx = (e^x)+C
4. ∫a^xdx = ((a^x)/(log_ea))+C
5. ∫sin(x)dx = -cos(x)+C
6. ∫cos(x)dx = sin(x)+C
7. ∫sec²(x)dx = tan(x)+C
8. ∫cosec²(x)dx = -cot(x)+C
9. ∫sec(x)tan(x)dx = sec(x)+C
10. ∫cosec(x)cot(x)dx = -cosec(x)+C
11. ∫cot(x)dx = log|sin(x)|+C
12. ∫tan(x)dx = log|sec(x)|+C
13. ∫sec(x)dx = log|sec(x)+tan(x)|+C
14. ∫cosec(x)dx = log|cosec(x)-cot(x)|+C

Definite Integrals

Definite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits.

\int_{a}^{b}F(x)dx, it denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit.

Note: If f is a continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then \int_{a}^{b}f(x)dx= \left [ F(x) \right ]_{a}^{b} = F(a) - F(b).

Here, the function f needs to be well defined and continuous in [a, b].

1. \int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt
2. \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx
3. \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx
4. \int_{a}^{b}f(x)=\int_{a}^{b}f(a+b-x)dx
5. \int_{0}^{b}f(x)=\int_{0}^{b}f(b-x)dx
6. \int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx+\int_{0}^{a}f(2a-x)dx
7. \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx, if f(x) is even function i.e f(x) = f(-x)
8. \int_{-a}^{a}f(x)dx=0, if f(x) is odd function

Newton-Leibnitz Rule

For a definite integral F(x) = \int_{a(x)}^{b(x)} f(t) \, dt:

\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} f(t) \, dt \right] = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)

Double Integral

The double integral of a function f(x, y) over a region R is denoted as:

∬_Rf(x, y) dA,

Where dA represents an infinitesimal area element, typically expressed as dx dy or dy dx.

If f(x, y) is continuous over R, the double integral represents the "accumulated value" of f(x, y) over the region R.

Geometric Interpretation

• Area: If f(x, y) = 1, the double integral computes the area of the region R.
  • Area = ∬_R 1 dA
• Volume: If f(x, y) represents the height above the xy-plane, the double integral computes the volume under the surface z = f(x, y) over the region R.

Triple Integral

The triple integral of a function f(x, y, z) over a three-dimensional region R is denoted as:

∭_R f(x, y, z) dV,

Where dV represents an infinitesimal volume element, typically expressed as dx dy dz

If f(x, y, z) = 1, the triple integral computes the volume of the region R:

Volume = ∭_R 1 dV.

Geometric Interpretation

A triple integral computes the "accumulated value" of f(x, y, z) over the three-dimensional region RRR. This can represent:

• Volume: If f(x, y, z) = 1, the integral computes the volume of R.
• Mass: If f(x, y, z) represents density, the integral computes the mass of the region RRR.

Application of Integrals

Some common application of integrals are:

• Area Under a Curve
  • Area Between Curves
  • Area Between Polar Curves
• Arc Length of Curves
• Volume of Solids of Revolution
• Surface Area of Revolution

Area Under a Curve

The area enclosed between a curve y = f(x), the x-axis, and the limits x = a and x = b is:

\text{Area} = \int_a^b f(x) \, dx

• If f(x) < 0, take the absolute value of the integral.

Between Two Curves

The area between two curves y = f(x) and y = g(x) from x = a to x = b is:

Area = \int_a^b \big| f(x) - g(x) \big| \, dx.

Length of a Curve

The length of a curve y = f(x) from x = a to x = b is:

Length = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx

For parametric equations x = x(t), y = y(t), the arc length is:

Length = \int_{t_1}^{t_2} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt

Volume of Solids of Revolution

• Disk Method: When a curve y = f(x) is revolved about the x-axis: Volume = \pi \int_a^b \left[ f(x) \right]^2 \, dx
• Shell Method: When a curve y = f(x) is revolved about the y-axis: Volume = 2\pi \int_a^b x \cdot f(x) \, dx.
• For parametric equations x = x(t), y = y(t):
  • Revolved about x-axis: Volume = \pi \int_{t_1}^{t_2} \left[ y(t) \right]^2 \frac{dx}{dt} \, dt

Surface Area of Solids of Revolution

1. Revolution about the x-axis:Surface Area = 2\pi \int_a^b f(x) \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
2. Revolution about the y-axis: Surface Area = 2\pi \int_a^b x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx

Taylor Series

The Taylor series of a function f(x) about a point x = a is given by:

f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \dots

In general:

f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n,

where:

• f^{(n)}(a) is the n^th derivative of f(x) evaluated at x = a,
• n! is n-factorial (n! = n ⋅ (n − 1)⋅ . . . ⋅ 1, with 0! = 1).

Maclaurin Series

The Maclaurin series is a special case of the Taylor series where a = 0:

f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \dots

In general:

f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}

Common Taylor Series Expansion

Some common expansion using taylor and maclaurin series are: