Comprehensive Calculus Exam Notes
1. Functions and Their Graphs
Definition (Formal): A rule f:A\rightarrow B that assigns every x\in A exactly one y=f(x)\in B.
• x – independent / input variable.
• y – dependent / output variable.
Domain & Range
• Domain = set of permissible x values.
• Range = set of attainable y values.
• Vertical–line test: a curve represents a function iff every vertical line intersects it at most once.
Common Restrictions
• Denominator \neq 0, radicand of even root \ge 0, argument of \log >0, etc.
Even / Odd / NENO (Neither-even-nor-odd)
• Even: f(-x)=f(x), graph symmetric about y-axis.
• Odd: f(-x)=-f(x), graph symmetric about origin.
• NENO: fails both tests.
Families of Functions
Algebraic – Polynomial, rational, irrational (root–type), modulus |x|, signum \operatorname{sgn}(x), GIF (floor) \lfloor x\rfloor, LIF (ceiling) \lceil x\rceil, fractional part {x}=x-\lfloor x\rfloor, piece-wise.
Transcendental – Exponential e^{x}, logarithmic \log_a x, trigonometric \sin x,\cos x,\tan x, inverse–trig, hyperbolic, etc.
Periodicity
f(x+T)=f(x)\;\forall x.
Examples: \sin x,\cos x have T=2\pi; \tan x has T=\pi.
Inverse Functions
Graphs of y=f(x) and y=f^{-1}(x) are mirror images in y=x.
Eg. y=e^{x} and y=\ln x.
2. Limits and Continuity
Neighbourhood of a
(a-h,\,a+h),\;h>0\text{ small}.
Limit ((x\to a))
\lim_{x\to a}f(x)=L if f(x) can be made arbitrarily close to L by taking x sufficiently close to a.
Left / right limits: \displaystyle \lim{x\to a^-},\;\lim{x\to a^+}.
Continuity at x=a
Function is continuous if
f(a) exists, 2. limit exists, 3. limit equals the value:
\lim_{x\to a}f(x)=f(a).
Common Discontinuities
Removable (hole), jump (piece-wise mismatch), infinite (vertical asymptote).
3. Indeterminate Forms & L’Hospital’s Rule
Seven classic indeterminate forms
\frac 0 0,\;\frac\infty\infty,\;0\cdot\infty,\;\infty-\infty,\;0^{0},\;\infty^{0},\;1^{\infty}.
If \displaystyle \lim{x\to a}\frac{f(x)}{g(x)} is \tfrac00 or \tfrac\infty\infty and f,g differentiable, then \displaystyle \lim{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} (differentiate numerator & denominator until resolved).
4. Differentiability and Derivatives
Derivative at a
f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h} if limit exists and is same from both sides.
• If derivative exists ⇒ function continuous.
• Converse not guaranteed (e.g., |x| continuous, not differentiable at 0).
Geometric meaning: slope of tangent; sharp points give unequal left/right derivatives ⇒ non-differentiable.
First-principle shortcuts
LHD = left limit of f'(x), RHD = right limit; equality is required.
5. Taylor & Maclaurin Series
For f differentiable to all orders near x=a,
f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots.
Maclaurin is the case a=0.
Useful Maclaurin expansions (|x| small):
• \ln(1+x)=x-\tfrac{x^{2}}{2}+\tfrac{x^{3}}{3}-\dots
• e^{x}=1+x+\tfrac{x^{2}}{2!}+\dots
• \sin x=x-\tfrac{x^{3}}{3!}+\tfrac{x^{5}}{5!}-\dots
• \cos x=1-\tfrac{x^{2}}{2!}+\tfrac{x^{4}}{4!}-\dots
Linear approximation: keep up to first power; quadratic approximation: keep up to (x-a)^2.
6. Mean Value Theorems
Rolle: If f continuous on [a,b], differentiable on (a,b) and f(a)=f(b) ⇒ \exists c\in(a,b):f'(c)=0.
Lagrange (MVT): Same hypotheses without f(a)=f(b) ⇒ \exists c such that
f'(c)=\dfrac{f(b)-f(a)}{b-a} (tangent parallel to chord).Cauchy MVT: For two functions f,g continuous on [a,b], differentiable on (a,b) and g'(x)\neq0, \exists c with
\dfrac{f'(c)}{g'(c)}=\dfrac{f(b)-f(a)}{g(b)-g(a)}.
Average (integral) MVT: If f continuous, \exists c such that
\dfrac1{b-a}\int_{a}^{b}f(x)dx=f(c).
7. Maxima and Minima
Critical / stationary point: f'(c)=0 or derivative undefined, but f continuous.
First-derivative test: sign change of f' around c.
Second-derivative test: if f''(c)\gt0 ⇒ local minimum, f''(c)\lt0 ⇒ local maximum; if f''(c)=0 use higher derivatives or first-test.
For two variables f(x,y) – Lagrange’s notation
r=f{xx},\;s=f{xy},\;t=f_{yy} at critical point.
Discriminant D=rt-s^{2}:
• D>0,\;r>0 ⇒ local min.
• D>0,\;r<0 ⇒ local max.
• D<0 ⇒ saddle (point of inflection).
• D=0 ⇒ inconclusive.
8. Partial, Total & Chain-Rule Derivatives
Partial derivatives:
f{x}=\partial f/\partial x\, (y\text{ const}), \quad f{y}=\partial f/\partial y\, (x\text{ const}).
Total derivative when x,y,z depend on t:
\dfrac{du}{dt}=f{x}\dfrac{dx}{dt}+f{y}\dfrac{dy}{dt}+f_{z}\dfrac{dz}{dt}.
Chain rule for functions of functions (two levels):
\frac{\partial u}{\partial r}=u{x}x{r}+u{y}y{r},\;\frac{\partial u}{\partial s}=u{x}x{s}+u{y}y{s}.
Jacobian of u,v w.r.t x,y:
J=\begin{vmatrix}u{x}&u{y}\v{x}&v{y}\end{vmatrix}.
If J=0 functions dependent; else independent.
Differential transform: dudv=|J|\,dxdy (and analogous in 3-D).
9. Integration – Essentials
Indefinite integral is antiderivative plus constant C.
Key formulas:
\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C\;(n\neq-1),\quad \int e^{x}dx=e^{x}+C,
\int \sin x\,dx=-\cos x+C,\quad \int \cos x\,dx=\sin x+C.
Integration techniques: substitution, parts \int udv=uv-\int vdu, partial fractions, trigonometric identities, rationalising, special transformations.
Definite integral & Fundamental Theorem
\displaystyle \int_{a}^{b}f(x)dx=F(b)-F(a) if F' = f.
Symmetry shortcuts (even/odd):
\int{-a}^{a}!f(x)dx=\begin{cases}2\int{0}^{a}f(x)dx,&f\text{ even}\0,&f\text{ odd}\end{cases}.
Induced limits:
If bounds depend on variable, Leibniz rule:
\dfrac{d}{dx}\int{\alpha(x)}^{\beta(x)}f(t,x)dt=f(\beta,x)\beta'(x)-f(\alpha,x)\alpha'(x)+\int{\alpha}^{\beta}\partial f/\partial x\,dt.
10. Beta & Gamma Functions
Gamma (second Euler integral):
\Gamma(n)=\int_{0}^{\infty} x^{n-1}e^{-x}\,dx,\quad n>0.
Special values: \Gamma(1)=1,\;\Gamma(n+1)=n!,\;\Gamma(\tfrac12)=\sqrt{\pi}.
Beta (first Euler integral):
B(m,n)=\int{0}^{1}x^{m-1}(1-x)^{n-1}dx=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}. Symmetry B(m,n)=B(n,m). Useful for evaluating integrals of type \int{0}^{\pi/2}\sin^{m}\theta\,\cos^{n}\theta\,d\theta.
11. Multiple Integrals & Jacobians
Double integral over region R: \iint_{R}f(x,y)\,dA; order can be dx\,dy or dy\,dx.
• Use vertical strip if y limits depend on x.
• Use horizontal strip if x limits depend on y.
Change of order requires redrawing region.
Triple integral: \iiint_{V}f(x,y,z)\,dV.
Coordinate transforms:
• Polar (2-D): x=r\cos\theta,\,y=r\sin\theta,\,dA=r\,dr\,d\theta.
• Cylindrical: x=r\cos\theta,\,y=r\sin\theta,\,z=z,\,dV=r\,dr\,d\theta\,dz.
• Spherical: x=\rho\sin\phi\cos\theta,\,y=\rho\sin\phi\sin\theta,\,z=\rho\cos\phi,
dV=\rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta.
12. Applications of Integration
Area (Cartesian): A=\displaystyle\int{x=a}^{b}\big[y{\text{top}}-y{\text{bottom}}\big]dx. In polar: A=\tfrac12\int{\theta1}^{\theta2}r^{2}d\theta.
Arc Length: For y=f(x), a\le x\le b
S=\displaystyle\int_{a}^{b}\sqrt{1+\left(f'(x)\right)^{2}}\,dx.
Parametric x(t),y(t):\;\displaystyle S=\int\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}dt.Volume of Revolution (about x-axis):
Disk method: V=\pi\int{a}^{b}\big[y(x)\big]^{2}dx. Shell method about y-axis: V=2\pi\int{a}^{b}x\,y(x)\,dx.Centre of Mass, Moment of Inertia: \iint_{R}\rho(x,y)\,x\,dA etc. (not detailed here).
13. Vector-Calculus Preview (Context)
Grad, div, curl, line/surface integrals, Green, Gauss, Stokes theorems connect differentiation & multiple integration; prepare by mastering Jacobians and coordinate transforms.
14. Quick Reference of Key Formulas
• Even test: f(-x)=f(x). Odd test: f(-x)=-f(x).
• L’Hospital: \displaystyle\lim\frac{f}{g}=\lim\frac{f'}{g'} (when applicable).
• Taylor (about a): f(x)=\sum\dfrac{f^{(n)}(a)}{n!}(x-a)^{n}.
• Mean slope: f'(c)=\dfrac{f(b)-f(a)}{b-a}.
• Arc length: \displaystyle\int\sqrt{1+(y')^{2}}dx.
• Beta–Gamma link: B(m,n)=\dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}.
• Polar area element: dA=r\,dr\,d\theta; cylindrical dV=r\,dr\,d\theta\,dz; spherical dV=\rho^{2}\sin\phi \, d\rho\,d\phi\,d\theta .
These consolidated notes integrate every principal definition, theorem, test, example and computational tool highlighted across the lecture transcript, providing a coherent single-source reference for rapid exam revision.