Calculus Study Notes: Core Concepts and Techniques
1. Limits and Continuity
A limit describes the value a function approaches as its input approaches a particular
value.
For a function (f(x)),
[ _{xa}f(x)=L ]
means that (f(x)) approaches (L) as (x) approaches (a).
Important techniques include:
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Direct substitution
Factoring and cancellation
Rationalization
Common denominators
Squeeze theorem
One-sided limits
A function is continuous at (a) when
[ _{xa}f(x)=f(a). ]
2. Derivatives
The derivative measures the instantaneous rate of change of a function.
The definition is
[ f’(x)=_{h}. ]
Important differentiation rules include:
[ xn=nx{n-1} ]
[ (f+g)‘=f’+g’ ]
[ (fg)‘=f’g+fg’ ]
[ ()’=. ]
The chain rule is
[ f(g(x))=f’(g(x))g’(x). ]
3. Applications of Derivatives
Critical points occur where (f’(x)=0) or where (f’(x)) does not exist.
The first derivative can be used to determine whether a function is increasing or
decreasing.
The second derivative provides information about concavity:
[ f’’(x)>0 ]
indicates concave upward behavior, while
[ f’’(x)<0 ]
indicates concave downward behavior.
The Mean Value Theorem states that if (f) is continuous on ([a,b]) and differentiable on
((a,b)), then some (c(a,b)) satisfies
[ f’(c)=. ]
4. Integrals
The definite integral can be interpreted as accumulated change or signed area.
The Fundamental Theorem of Calculus connects differentiation and integration:
[ _a^x f(t),dt=f(x). ]
If (F’(x)=f(x)), then
[ _a^b f(x),dx=F(b)-F(a). ]
Common integration techniques include:
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Substitution
Integration by parts
Trigonometric integrals
Trigonometric substitution
Partial fractions
Numerical integration
5. Sequences and Series
A sequence is a function whose domain is the positive integers.
An infinite series has the form
[ _{n=1}^{}a_n. ]
Important convergence tests include:
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Divergence test
Integral test
Comparison test
Limit comparison test
Alternating series test
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Ratio test
Root test
The geometric series
[ _{n=0}{}arn ]
converges when (|r|<1), with sum
[.]
6. Parametric and Polar Equations
A parametric curve is represented by
[ x=f(t),y=g(t). ]
Its derivative is
[ =. ]
Polar coordinates represent points using
[ x=r,y=r. ]
The area enclosed by a polar curve can be computed using
[ A=_r2,d. ]
Multivariable Calculus
7. Vectors and Three-Dimensional Geometry
A vector in (R^3) can be represented as
[ v=a,b,c. ]
Its magnitude is
[ |v|=. ]
The dot product is
[ uv =u_1v_1+u_2v_2+u_3v_3. ]
The cross product produces a vector perpendicular to both input vectors.
8. Vector-Valued Functions
A vector-valued function has the form
[ r(t)=x(t),y(t),z(t). ]
Its derivative is computed componentwise:
[ r’(t) =x’(t),y’(t),z’(t). ]
The derivative represents the tangent vector to the curve.
9. Partial Derivatives
For a function
[ f(x,y), ]
the partial derivative with respect to (x) is calculated while treating (y) as constant:
[ f_x=. ]
Similarly,
[ f_y=. ]
The gradient is
[ f =f_x,f_y. ]
For three variables,
[ f =f_x,f_y,f_z. ]
10. Tangent Planes and Linearization
For (z=f(x,y)), the tangent plane at ((a,b)) is
[ z-f(a,b) =f_x(a,b)(x-a)+f_y(a,b)(y-b). ]
The linearization is therefore
[ L(x,y) =f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b). ]
11. Multiple Integrals
A double integral over a region (D) is written
[ _D f(x,y),dA. ]
In rectangular coordinates,
[ _D f(x,y),dA = a^b{g_1(x)}^{g_2(x)} f(x,y),dy,dx. ]
Triple integrals extend this idea to three dimensions:
[ _E f(x,y,z),dV. ]
12. Vector Calculus
A vector field in three dimensions can be written
[ F(x,y,z) =P(x,y,z),Q(x,y,z),R(x,y,z). ]
The divergence is
[ F =P_x+Q_y+R_z. ]
The curl is
[ F. ]
Important theorems connecting vector fields, line integrals, surface integrals, and volume
integrals include:
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Green’s Theorem
Stokes’ Theorem
Divergence Theorem
These provide different ways to transform difficult integrals into equivalent but easier ones.
Study Strategy
For each section, first attempt problems without looking at solutions. If an answer is
incorrect, identify the first step where the reasoning diverged from the correct solution.
For computational exercises, focus on understanding the method rather than memorizing
individual answers.
For proof-oriented or conceptual problems, write a complete explanation before consulting
a solution.
A useful progression is:
1. Read the section.
2. Work through the examples.
3. Attempt exercises independently.
4. Check difficult problems against a solution manual.
5. Rework missed problems without looking at the solution.
6. Summarize the underlying technique in your own words.