High School Calculus

Limits • Derivatives • Integrals • Applications

Grade 12 | Truth Carriers Academy

Study Guide (Read First)

Goal: Understand how limits lead to derivatives and integrals, master key rules, and apply calculus to real problems while recognizing the mathematical order Yahuah built into creation.

“He has made the earth by His power, He has established the world by His wisdom, and stretched out the heavens by His understanding.” - Jeremiah 51:15

Table of Contents

1Limits & Continuity

Core Idea

Limits describe the value a function approaches as x approaches a point.

limx→a f(x) = L

Check Understanding

Evaluate lim(x→2) (x² - 4)/(x-2) =

Describe in words what a limit means:

2Derivatives: Definition & Rules

Definition

f'(x) = limh→0 [f(x+h) - f(x)] / h

Derivative = instantaneous rate of change = slope of tangent line.

Basic Rules

Try It

Find d/dx (3x³ - 5x + 7) =

Find d/dx (sin(3x²)) using chain rule:

3Applications of Derivatives

Key Uses

Scenario

A particle has s(t) = t³ - 6t² + 9t. Find when it is at rest (v=0):

4Exponential & Logarithmic Functions

Derivatives

Try It

d/dx (5e^x) =

d/dx (ln(3x)) =

5Trigonometric Derivatives

Core Derivatives

FunctionDerivative
sin xcos x
cos x-sin x
tan xsec² x
sec xsec x tan x

Try It

d/dx (sin(2x)) =

6Implicit Differentiation & Related Rates

Implicit Steps

  1. Differentiate both sides
  2. Apply chain rule to y terms (multiply by y')
  3. Solve for y'

Implicit Example

For x² + y² = 25, find dy/dx:

Related Rates

A balloon rises at 3 ft/s. You stand 40 ft away. How fast is the distance changing when the balloon is 30 ft high? (use Pythagorean then differentiate):

7Integrals & Antiderivatives

Antiderivatives

If F'(x) = f(x), then F is an antiderivative of f.

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

Other basics: ∫ e^x dx = e^x + C, ∫ 1/x dx = ln|x| + C

Definite Integral

Represents signed area under a curve from a to b.

∫ab f(x) dx

Try It

∫ (6x² - 4) dx =

8Fundamental Theorem of Calculus

The Bridge

Part 1: If F(x) = ∫ax f(t) dt, then F'(x) = f(x).

Part 2: ∫ab f(x) dx = F(b) - F(a) where F' = f.

FTC shows differentiation and integration are inverse processes. This elegant unity testifies to the order built into creation.

Apply It

Compute ∫02 (3x²) dx using FTC:

9Applications of Integration

Common Applications

Try It

Area between y = x² and y = 2x on [0,2]:

10Sequences & Series (Intro)

Key Ideas

Check

Does ∑n=0∞ (1/2)ⁿ converge? Sum =

11Intro to Differential Equations

Basic Idea

A differential equation relates a function to its derivatives. Solutions are functions, not numbers.

Try It

Solve y' = 3y with y(0)=2:

12Review & Worldview

Key Takeaways

“Great are the works of Yahuah; they are studied by all who delight in them.” - Psalm 111:2

Final Reflection

How does the unity of calculus (FTC) reinforce your view of Yahuah’s order in creation?

Answer Key (Selected)

Unit 1: Limit = 4 (factor to (x+2));

Unit 2: d/dx (3x³ - 5x +7) = 9x² -5

Unit 4: d/dx (5e^x) = 5e^x

Unit 5: d/dx sin(2x) = 2cos(2x)

Unit 7: ∫ (6x² -4) dx = 2x³ -4x + C

Unit 8: ∫₀² 3x² dx = [x³]₀² = 8

Unit 10: Geometric sum = 1/(1-1/2) = 2

Unit 11: y = 2e^{3t}