# CALC Endterm Study Plan (Based on Endterm 2025)

## Tier 1: Very High frequency + on Endterm 2025 -- study these first

| # | Topic | 2025 Endterm Q | Lectures | Pattern | Done |
|---|-------|---------------|----------|---------|------|
| 1 | Power Series: Interval / Radius of Convergence | O1a | L13 | P1 | [x] |
| 2 | Critical Points + Second Derivative Test (Multivariable) | O2a | L19 | P2 | [x] |
| 3 | Double Integrals (setup, evaluate, interchange order) | O3 | L22, L23 | P3 | [x] |
| 4 | Complex Roots (z^n = c) | S3 | L20, L21 | P4 | [x] |
| 5 | Linearization (from power series) | O1b | L04, L14 | P5 | [x] |
| 6 | Directional Derivative | S1a, S1b | L17, L18 | P6 | [x] |
| 7 | Improper Integrals | S4 | L09 | P7 | [x] |

## Tier 2: High frequency + on Endterm 2025

| # | Topic | 2025 Endterm Q | Lectures | Pattern | Done |
|---|-------|---------------|----------|---------|------|
| 8 | Taylor Series for Anti-Derivative + Error Bound | O4a, O4b | L15 | P13 | [x] |
| 9 | Absolute Max/Min on Closed Domain (Multivariable) | O2b | L19 | P14 | [x] |
| 10 | Series Convergence / Divergence Classification | S5, O7 | L12 | P16 | [ ] |
| 11 | Higher-Order Derivative from Power Series | S2 | L14 | P20 | [x] |

## Tier 3: On Endterm 2025 but low frequency

| # | Topic | 2025 Endterm Q | Lectures | Pattern | Done |
|---|-------|---------------|----------|---------|------|
| 12 | Euler's Formula / Complex Trig Identities | O6 | L21 | P30 | [ ] |
| 13 | Newton-Raphson Method | O5 | L05 | P33 | [ ] |

## Tier 4: Very High frequency but NOT on Endterm 2025 -- review if time permits

| # | Topic | Lectures | Pattern | Done |
|---|-------|----------|---------|------|
| 14 | Recursive Sequences (monotonicity, boundedness, limit) | L10 | P8 | [ ] |
| 15 | Implicit Differentiation | L05 | P9 | [ ] |
| 16 | Gradient Vector | L17, L18 | P15 | [ ] |
| 17 | Match Functions to Graphs / Contour Plots | L16 | P17 | [ ] |
| 18 | Complex Number Arithmetic (a+bi form) | L20 | P18 | [ ] |
| 19 | Geometric / Telescoping Series (Find Sum) | L11 | P21 | [ ] |
| 20 | Limits via Taylor Series Expansion | L15 | P28 | [ ] |

---

## Endterm 2025 Questions

| Q# | Topic | Pattern | Lectures | Done |
|----|-------|---------|----------|------|
| S1a | Directional derivative: give formula for h(x,y) = D_u g(x,y) where u = <sqrt2/2, -sqrt2/2> | P6 | L17, L18 | [x] |
| S1b | Directional derivative of h(x,y) in direction v = <1,1> at (1,2) | P6 | L17, L18 | [x] |
| S2 | f(x) = x^2 arctan(x), compute f^(9)(0) via Taylor series | P20 | L14 | [x] |
| S3 | Solve (z-2)^3 = -8, give in a+bi form | P4 | L20, L21 | [x] |
| S4 | Compute integral from 0 to inf of e^{-sqrt(x)} dx | P7 | L09 | [x] |
| S5 | Series convergence test for sum (-1)^n * n/sqrt(n^2-3) -- identify correct test | P16 | L12 | [x] |
| O1a | Interval of convergence for sum ln(n)/n * (x/2 - 1)^n | P1 | L13 | [x] |
| O1b | Linearization of g(x) at center of interval of convergence | P5 | L04, L14 | [x] |
| O2a | Critical points of h(x,y) = 1-34x-12y+8xy+8x^2 in triangle (1,1),(2,1),(2,2), classify | P2 | L19 | [x] |
| O2b | Absolute max and min of h on the triangle | P14 | L19 | [x] |
| O3 | Double integral of xy over region in first quadrant (left of y=5-x, y<=4, excluding triangles) | P3 | L22, L23 | [ ] |
| O4a | Taylor series centered at 0 for anti-derivative of f(x) = e^{x^2} | P13 | L15 | [x] |
| O4b | Approximate integral from 0 to 1 of f(x) dx with error below 0.06 | P13 | L15 | [x] |
| O5 | Newton-Raphson sketch for cube root of 6 | P33 | L05 | [ ] |
| O6 | Prove e^{i*theta} = cos(theta) + i*sin(theta) using power series | P30 | L21 | [ ] |
| O7 | Classify: (a) sum (-1)^n n^2 e^{-n}, (b) sum (1+sin(n))/n^2 | P16 | L12 | [x] |

---

## Resit 2025 Questions

| Q# | Topic | Lectures | Done |
|----|------|---------|-----|
| S1 | Limit with root trick / L'Hopital | L01 | [x] |
| S2 | Asymptotes (vertical, horizontal, oblique) | L03 | [x] |
| S3 | Implicit differentiation for level curve slope | L05 | [x] |
| S4 | Integration by parts (iterated) | L07 | [x] |
| S5 | Contour plot matching | L16 | [x] |
| S6 | Convergence: geometric series, improper integrals, integral test | L09, L11, L12 | [x] |
| O1a | Power series interval of convergence (geometric decomposition) | L11, L13 | [ ] |
| O1b | Power series represents function (geometric series identity) | L11, L13 | [ ] |
| O2a | Critical points of multivariable, classify | L19 | [ ] |
| O2b | Absolute max/min on triangular region | L19 | [ ] |
| O3 | Double integral over region (type II, x-simple) | L22, L23 | [ ] |

---

## How to Solve Each Pattern

### P1 -- Power Series: Interval / Radius of Convergence (5 pts)
**How to recognize:** "Find the interval of convergence of sum c_n (x-a)^n"

**Steps:**
1. **Ratio test:** L = lim |c_{n+1}/c_n|, then R = 1/L
2. **Root test (if c_n has n-th powers):** L = lim |c_n|^{1/n}, then R = 1/L
3. **Non-standard forms:** if series has x^{2n}, substitute u = x^2, find R for u, convert back
4. **Boundary check:** substitute x = a+R and x = a-R into the series, test each separately:
   - Alternating series test, p-series, comparison test, limit comparison
5. **Write the interval:** e.g., [a-R, a+R) if left endpoint converges and right diverges

**2025 exam:** sum ln(n)/n * (x/2-1)^n. Let u = x/2-1, center at u=0 (x=2). Ratio/root test to find R. Check endpoints.

### P2 -- Critical Points + Second Derivative Test (Multivariable) (3 pts)
**How to recognize:** "Find critical points of f(x,y) in the interior of domain D, classify"

**Steps:**
1. Compute f_x and f_y
2. Solve f_x = 0 AND f_y = 0 simultaneously
3. Check which solutions lie inside the domain
4. Compute D = f_xx * f_yy - (f_xy)^2 at each critical point
   - D > 0 and f_xx > 0 -> local min
   - D > 0 and f_xx < 0 -> local max
   - D < 0 -> saddle
   - D = 0 -> inconclusive

### P3 -- Double Integrals (5 pts)
**How to recognize:** "Compute the double integral of f(x,y) dA over region D"

**Steps:**
1. **Sketch the region** -- draw all boundary curves, find intersection points
2. **Choose integration order** -- pick the order with simpler limits
   - If integrand is hard to integrate in one variable (e.g., cos(x^2)), try the other order
3. **Set up limits** -- for dy dx: x goes a to b, y goes g1(x) to g2(x)
4. **Split if needed** -- if boundary changes form, split into sub-regions
5. **Evaluate** -- inner integral first (treat outer variable as constant)

**2025 exam tip:** Region is irregular (first quadrant, left of y=5-x, y<=4, excluding triangles). Sketch carefully, split into sub-regions if needed.

### P4 -- Complex Roots (3 pts)
**How to recognize:** "Find all complex solutions to z^n = c" or "(z-a)^n = c"

**Steps:**
1. Convert c to polar: c = r * e^{i*theta}
2. z_k = r^{1/n} * e^{i*(theta + 2*pi*k)/n} for k = 0, 1, ..., n-1
3. Convert to a+bi: z_k = r^{1/n} * (cos(phi_k) + i*sin(phi_k))
4. For (z-a)^n = c: solve w^n = c first, then z = w + a

**2025 exam:** (z-2)^3 = -8. Set w = z-2. -8 = 8*e^{i*pi}. w_k = 2*e^{i*(pi+2*pi*k)/3}.
- k=0: 2*e^{i*pi/3} = 1 + i*sqrt(3)
- k=1: 2*e^{i*pi} = -2
- k=2: 2*e^{i*5pi/3} = 1 - i*sqrt(3)
- Then z_k = w_k + 2: z = 3+i*sqrt(3), z = 0, z = 3-i*sqrt(3)

### P5 -- Linearization (1 pt)
**How to recognize:** "Find the linearization of f at x=a" or "approximate f(value) using linearization"

**Formula:** L(x) = f(a) + f'(a)(x - a)

**From power series:** If g(x) = sum c_n(x-a)^n, then linearization at x=a is just L(x) = c_0 + c_1(x-a).

### P6 -- Directional Derivative (4 pts)
**How to recognize:** "Find D_u f at point P" or "direction of steepest ascent/descent"

**Steps:**
1. Compute gradient: nabla f = (f_x, f_y)
2. Normalize direction: u_hat = u / |u|
3. D_u f = nabla f . u_hat
4. Max rate of change = |nabla f|, in direction nabla f / |nabla f|
5. Min rate of change = -|nabla f|, in direction -nabla f / |nabla f|

**2025 exam twist:** S1a asks for h(x,y) = D_u g(x,y) as a formula (not evaluated at a point). Then S1b asks for the directional derivative of h itself. Just compute nabla g, dot with u to get h(x,y) as a function, then compute nabla h and dot with v.

### P7 -- Improper Integrals (3 pts)
**How to recognize:** Integral with infinity limit or singularity in integrand

**Steps:**
- Type 1 (inf limit): replace inf with t, evaluate, take lim t->inf
- Type 2 (singularity): split at singularity, use limit
- Common techniques: substitution, IBP, comparison test, p-test
- p-test: int_1^inf 1/x^p converges iff p > 1

**2025 exam:** int_0^inf e^{-sqrt(x)} dx. Substitution u = sqrt(x), x = u^2, dx = 2u du. Becomes int_0^inf 2u*e^{-u} du. IBP: = 2[-u*e^{-u} - e^{-u}] from 0 to inf = 2(0 - (-1)) = 2.

### P13 -- Taylor Series for Anti-Derivative + Error Bound (6 pts)
**How to recognize:** "Give the Taylor series for an anti-derivative of f(x)" then "approximate integral with error below epsilon"

**Steps:**
1. Start from known series (e^x = sum x^n/n!, etc.)
2. Substitute to get series for f(x). E.g., f(x) = e^{x^2} -> sum (x^2)^n/n! = sum x^{2n}/n!
3. Integrate term by term: F(x) = C + sum x^{2n+1}/((2n+1)*n!)
4. For definite integral: int_0^b f(x) dx = sum b^{2n+1}/((2n+1)*n!)
5. Error bound (alternating): |error| <= |first omitted term|
6. Error bound (positive): compare remainder with geometric tail or use integral estimate
7. Sum terms until the bound is met

**2025 exam:** f(x) = e^{x^2}. Series: sum x^{2n}/n!. Anti-derivative: sum x^{2n+1}/((2n+1)*n!). Approximate int_0^1 with error < 0.06: sum 1/((2n+1)*n!) until remainder < 0.06.

### P14 -- Absolute Max/Min on Closed Domain (Multivariable) (3 pts)
**How to recognize:** "Find the absolute maximum and minimum of f on D"

**Steps:**
1. Find interior critical points (from P2 above), evaluate f at each
2. Parametrize each boundary edge:
   - For triangle (1,1)-(2,1)-(2,2): three edges, each becomes a single-variable problem
   - Edge 1: y=1, 1<=x<=2. Edge 2: x=2, 1<=y<=2. Edge 3: y=x, 1<=x<=2
3. Find critical points on each edge (set derivative = 0), evaluate f
4. Evaluate f at all corners/vertices
5. Compare all values -> largest = abs max, smallest = abs min

### P16 -- Series Convergence / Divergence Classification (6 pts)
**How to recognize:** "Determine if absolutely convergent, conditionally convergent, or divergent"

**Decision tree:**
1. **Divergence test FIRST:** if lim a_n != 0, DIVERGES. Done.
2. **Check absolute convergence:** test sum |a_n|
   - Ratio test: lim |a_{n+1}/a_n| < 1 -> abs convergent
   - Root test: lim |a_n|^{1/n} < 1 -> abs convergent
   - Comparison with p-series or geometric
3. **If sum |a_n| diverges, check conditional:** use alternating series test (b_n decreasing, lim b_n = 0)

**2025 exam:**
- S5: sum (-1)^n * n/sqrt(n^2-3). lim n/sqrt(n^2-3) = 1 != 0. DIVERGES by divergence test.
- O7a: sum (-1)^n n^2 e^{-n}. Ratio test on |a_n| = n^2/e^n: lim = 1/e < 1. Absolutely convergent.
- O7b: sum (1+sin(n))/n^2. Since 0 <= 1+sin(n) <= 2, we have 0 <= a_n <= 2/n^2. Comparison with 2/n^2 (convergent p-series). Converges (absolutely, all terms non-negative).

### P20 -- Higher-Order Derivative from Power Series (3 pts)
**How to recognize:** "Compute f^(k)(0)" where f involves a product with arctan, sin, etc.

**Key identity:** if f(x) = sum c_n x^n, then f^(k)(0) = k! * c_k

**Steps:**
1. Find the Taylor series of the inner function (e.g., arctan(x) = sum (-1)^n x^{2n+1}/(2n+1))
2. Multiply by the polynomial factor (e.g., x^2 * arctan(x) = sum (-1)^n x^{2n+3}/(2n+1))
3. Find the coefficient c_k of x^k
4. f^(k)(0) = k! * c_k

**2025 exam:** f(x) = x^2 arctan(x). arctan(x) = sum (-1)^n x^{2n+1}/(2n+1). So f(x) = sum (-1)^n x^{2n+3}/(2n+1). The x^9 term: 2n+3=9 -> n=3. c_9 = (-1)^3/7 = -1/7. f^(9)(0) = 9! * (-1/7) = -51840.

### P30 -- Euler's Formula (2 pts)
**How to recognize:** "Show/prove that e^{i*theta} = cos(theta) + i*sin(theta)"

**Proof:**
1. e^{i*theta} = sum (i*theta)^n / n!
2. Separate even and odd terms:
   - Even: sum (-1)^k theta^{2k} / (2k)! = cos(theta)
   - Odd: i * sum (-1)^k theta^{2k+1} / (2k+1)! = i*sin(theta)
3. Therefore e^{i*theta} = cos(theta) + i*sin(theta)

### P33 -- Newton-Raphson (1 pt)
**How to recognize:** "Sketch Newton-Raphson converging to..."

**Steps:**
1. Choose f(x) such that root = desired value. For cube_root(6): f(x) = x^3 - 6
2. Draw the curve y = f(x)
3. Pick x_0 (e.g., x_0 = 2 since 2^3=8 is close)
4. Draw tangent at (x_0, f(x_0)), mark where it hits x-axis -> that's x_1
5. Repeat: tangent at x_1 -> x_2. Show convergence toward the root

---

## Points Distribution Summary

| Category | Points | % of exam |
|----------|--------|-----------|
| Power Series + Taylor (P1 + P13 + P20) | 14 | 30% |
| Multivariable Calculus (P2 + P3 + P6 + P14) | 15 | 33% |
| Series Convergence (P16) | 6 | 13% |
| Complex Numbers (P4 + P30) | 5 | 11% |
| Improper Integrals (P7) | 3 | 7% |
| Other (P5, P33) | 2 | 4% |
| **Total** | **45** | **100%** |

**Priority: Multivariable (33%) and Power Series/Taylor (30%) together cover 63% of the exam.**