Visual differential calculus

This expository paper is intended to provide engineering and technology students with a purely visual and intuitive approach to differential calculus. The plan is that students who see intuitively the benefits of the strategies of calculus will be encouraged to master the algebraic form changing techniques such as solving, factoring and completing the square. Differential calculus will be treated as a continuation of the study of branches¹¹ of continuous and smooth curves described by equations which was initiated in a pre-calculus or advanced algebra course. Functions are defined as the single valued expressions which describe the branches of the curves. Derivatives are secondary functions derived from the just mentioned functions in order to obtain the slopes of the lines tangent to the curves. The derivatives of the derivatives are related to the turning of the tangent lines. The concepts involved in the study of continuous curves are not difficult to comprehend. But parsing complexity arises from the many combinations of forms, kinds and operations on functions. The paper will focus on the properties of continuous and smooth curves while maintaining an organized topic structure. Subsequently a student, enabled with the goals and structure of a course in differential calculus, can refer to conventional texts to fill in and expand on subordinate details.