Resource 1: Calculus: Early Transcendentals

I would recommend starting your preparations with this book: 

• Title: Calculus: Early Transcendentals 

• Author: James Stewart

Suggested Study Outline

I would recommend working the following chapters and sections, listed from the 6th edition. 

Chapter 10: Parametric Equations and Polar Coordinates

• 10.1: Curves Defined by Parametric Equations

• 10.2: Calculus with Parametric Curves

• 10.3: Polar Coordinates

• 10.4: Areas and Lengths in Polar Coordinates

• 10.5: Conic Sections

• 10.6: Conic Sections in Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

• 12.1: Three-Dimensional Coordinate Systems

• 12.2: Vectors

• 12.3: The Dot Product

• 12.4: The Cross Product

• 12.5: Equations of Lines and Planes

• 12.6: Cylinders and Quadric Surfaces

Chapter 13: Vector Functions

• 13.1: Vector Functions and Space Curves

• 13.2: Derivatives and Integrals of Vector Functions

• 13.3: Arc Length and Curvature

• 13.4: Motion in Space: Velocity and Acceleration

Remarks

• Pay close attention to 10.5 and 10.6, which introduce conic sections.

• At the very end of 13.4 you will find a discussion about Kepler's Laws of planetary motion.

• As you may have guessed, this is the point at which we make the connection between conic sections, Kepler's Laws and Newton's Law of Universal Gravitation.

• The Stewart text does not work through the proofs; instead, it offers hints and leaves it as an exercise to the reader. Not to worry, there are plenty more resources which do show the proof. The important thing is that if you have done all the work up to 13.4, you will be sufficiently armed to follow along the derivations.

• Most undergraduate calculus books seem to have a section discussing or proving Kepler's laws; in one of them, I found this tiny footnote hidden away in a corner, which I think should have been written in huge capital letters:

"If some of the steps seem unanticipated to you, you should realize that we are discussing one of the most celebrated physical problems in history and there have been three hundred years to think of ingenious ways to deal with it." 

This quote is from Calculus: One and Several Variables by Salas, Hille and Etgen, Section 14.7 (Planetary Motion).