The derivative of a function at a point tells us the "steepness" or instantaneous rate of change of the function at that point. We start by considering a secant line connecting two points on the curve: (x, f(x)) and (x+h, f(x+h)). The slope of this secant line is (f(x+h) - f(x)) / h.
As we make h (the horizontal distance between the points) smaller and smaller, the secant line approaches the tangent line to the curve at x. The slope of this tangent line is the derivative, f'(x).
Secant Slope: N/A | Approx. Tangent Slope (f'(x)): N/A
y = f(x) is plotted.P1 = (x_val, f(x_val)) and P2 = (x_val + h_val, f(x_val + h_val)).m_secant = (f(x_val + h_val) - f(x_val)) / h_val.h_val approaches 0, the secant line becomes the tangent line. Its slope is the derivative. We approximate this with a very small h for visualization.The definite integral of a function f(x) from a to b represents the accumulated area between the curve and the x-axis over that interval. We can approximate this area using Riemann Sums. This involves dividing the interval [a, b] into N smaller subintervals (rectangles).
The width of each rectangle is dx = (b-a)/N. The height can be determined by the function's value at the left endpoint, right endpoint, or midpoint of each subinterval. As N (the number of rectangles) increases, the sum of the areas of these rectangles gets closer to the true area under the curve.
Approx. Area (Riemann Sum): N/A
y = f(x) is plotted.[a, b] is divided into N subintervals, each of width dx = (b-a)/N.x_i = a + i * dxf(x_i)f(x_i) * dxSum(Area_i) for i from 0 to N-1.N -> infinity.A vector field assigns a vector (an arrow with direction and magnitude) to every point in a 2D plane or 3D space. It's defined by component functions, e.g., F(x,y) = . P(x,y) gives the x-component of the vector at point (x,y), and Q(x,y) gives the y-component.
Vector fields can represent physical phenomena like wind patterns, fluid flow, or force fields (like gravity or electromagnetism). We visualize them by drawing arrows at various grid points.
(x,y) on a grid, we calculate vx = P(x,y) and vy = Q(x,y).(x,y) and ending near (x + scale*vx, y + scale*vy). The 'scale' factor is for visual clarity.P(x,y) = -y, Q(x,y) = x (creates circular flow).P(x,y) = x, Q(x,y) = y (vectors point away from origin).P(x,y) = 1, Q(x,y) = 0 (all vectors point right).