Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit.
Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the [latex]x[/latex]-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.
Key Equations
Slope of a Secant Line
[latex]m_{\sec}=\dfrac{f(x)-f(a)}{x-a}[/latex]
Average Velocity over Interval [latex][a,t][/latex]
[latex]v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}[/latex]
