This text is designed for the usual introductory physics courses to prepare undergraduate students for the level of mathematics expected in more advanced undergraduate physics and engineering courses. One of its goals is to guide the student in learning the mathematical language physicists use by leading them through worked examples and then practicing problems. The pedagogy is that of introducing concepts, designing and refining methods, and practicing them repeatedly in physics examples and problems. Geometric and algebraic approaches and methods are included and are more or less emphasized in a variety of settings to accommodate different learning styles of students. Sometimes examples are solved in more than one way. Theorems are usually derived sketching the underlying ideas and describing the relevant mathematical relations so that one can recognize the assumptions they are based on and their limitations. These proofs are not rigorous in the sense of the professional mathematician, and no attempt was made to formulate theorems in their most general form or under the least restrictive assumptions.