📐 Derivatives at a Point
MATH → 8: nDeriv(
Format: nDeriv(expression, variable, point)
Example: Find derivative of x² at x=3
nDeriv(X², X, 3) → 6
For trigonometric functions:
nDeriv(sin(X), X, π/2) → 0
⚠️ Important: nDeriv uses numerical approximation (h = 0.001). For exact symbolic derivatives, use calculus.
💡 Pro Tip: Store the point in a variable first for repeated calculations:
3 → X
nDeriv(X², X, X) → 6
∫ Definite Integrals
MATH → 9: fnInt(
Format: fnInt(expression, variable, lower, upper)
Example: ∫₀² x² dx
fnInt(X², X, 0, 2) → 2.666666667
For improper integrals:
fnInt(e^(-X²), X, -1E99, 1E99) → 1.772453851
(Approximates ∫₋∞^∞ e⁻ˣ² dx = √π)
💡 Checking Area: Graph the function first, then use:
2nd → TRACE → 7: ∫f(x)dx
Set lower and upper bounds graphically.
🔍 Equation Solver
MATH → 0: Solver...
Solve equations of the form expression = 0
1. Press MATH
2. Select 0: Solver...
3. Enter equation (e.g., X³ - 3X + 1 = 0)
4. Press ALPHA → SOLVE
Example: Find roots of x³ - 3x + 1 = 0
Enter: X³ - 3X + 1 = 0
Guess: 0 → Answer: 0.3472963553
⚠️ Multiple Solutions: Try different initial guesses to find all roots.
💡 Critical Points: Set derivative = 0 to find local extrema.
📈 Graph Analysis Tools
After graphing function(s) in Y= menu:
2nd → TRACE → CALC Menu
- 1: value - Evaluate function at specific x
- 2: zero - Find x-intercepts (roots)
- 3: minimum - Find local minimum
- 4: maximum - Find local maximum
- 5: intersect - Find intersection of two curves
- 6: dy/dx - Find derivative at a point
- 7: ∫f(x)dx - Find definite integral
Finding Maximum:
1. Graph function
2. 2nd → TRACE → 4:maximum
3. Set left/right bounds around peak
4. Press ENTER for maximum coordinates
∞ Limits & Sequences
Limits using tables:
1. Press 2nd + WINDOW (TBLSET)
2. Set TblStart near limit point
3. Set ΔTbl = 0.001 for accuracy
4. Press 2nd + GRAPH (TABLE)
Example: limₓ→₀ (sin x)/x
TblStart = -0.01, ΔTbl = 0.001
Table shows values approaching 1
Summation (Riemann sums):
List → MATH → 5: sum(
sum(seq(f(X), X, a, b, Δx))
Right Riemann sum for ∫₀² x² dx with n=4:
sum(seq(0.5*(0.5X)², X, 1, 4, 1)) → 3.75
📊 Table of Values Setup
2nd → WINDOW (TBLSET)
Key Settings:
- TblStart: Starting x-value
- ΔTbl: Increment between x-values
- Indpnt: Auto (automatic x-values)
- Depend: Auto (automatic y-values)
For derivative approximation:
TblStart = point - 0.01
ΔTbl = 0.001
View table to see f(x) values approaching limit
💡 Quick Table: After graphing, press 2nd + GRAPH to see table of values at current x settings.
🔄 Function Operations
Store and recall functions:
Store: X² - 3X → Y₁
Recall: Y₁(2) evaluates at x=2
Chain calculations:
Find derivative and integral of same function:
X³ - 2X → Y₁
nDeriv(Y₁, X, 1) → 1
fnInt(Y₁, X, 0, 1) → -0.75
Graph derivatives:
Y₁ = original function
Y₂ = nDeriv(Y₁, X, X)
Graph both to see f(x) and f'(x)
🎯 AP Calculus Exam Tips
Exam Mode: Some exams require clearing memory. Know how to reset:
2nd → + → 7:Reset → 1:All Ram → 2:Reset
Time Savers:
- Store frequently used numbers (π, e) in variables
- Use ANS key for previous answer in next calculation
- Learn shortcut menus: 2nd + SIN for sin⁻¹, etc.
- Use parentheses liberally to ensure correct order of operations
Quick π calculations:
π → P
sin(P/2) → 1