Calculus you are expected to know for Phys 113Graphical Meaning of Derivatives and IntegralsThe derivative of f(x) with respect to x gives the slope of f(x) as a function of x: )(xfdxd[Slope of f(x) (x)]This means that if you take a derivative of f(x) with respect to x, and evaluate it at a specific x x-value you'll get the slope of f(x) at that specific x-value.The integral of f(x) with respect to x gives the area under the f(x) curve as a function of x:dxxfdxd)([Area under f(x) (x)]This means that if you take an integral of f(x) with respect to x, you'll get the area under the curve up to a general x. If you evaluate that at a specific x-value, you'll get the area under the curve up to that specific x-value.The problem here (and the reason that you need to add a constant when you take an indefinite integral) is that you don't know where to START adding the area under the curve. If you do not add a constant, then you effectively decided to start adding at x=0. The constant represents whatever area came before x = 0....Basic Differentiation and Integration of Polynomials, Sine, Cosine, and Exponential Functions The following was taken from Appendix E of Halliday Resnick and Walker's Fundamentals of Physics, 6th ed. (It's a little fuzzy... but essentially the same thing is in your own book's