explain about limit and first principle and how it helps to explain instantaneous speed .accelration etc ---**Textual Information:** * "Not very good" * "f(x)" * "m = Δy / Δx" **Chart/Diagram Description:** * **Type:** Cartesian coordinate system with a graph of a function and a secant line segment. * **Coordinate Axes:** X-axis and Y-axis are labeled. The origin is marked as 0. The Y-axis has scale markings at 2, 4, 6, 8. The X-axis has scale markings, with specific points labeled x₁ and x₂. * **Curved Line:** Represents a function labeled f(x). The curve appears to start near y=1 on the y-axis and curves upwards. * **Points:** Two distinct points on the curve f(x) are marked with white circles. * The first point is labeled (x₁, y₁) and is located at x = x₁ and y = y₁. A horizontal dashed line extends from this point to the y-axis (not explicitly labeled on the axis) and a vertical dashed line extends to the x-axis at x₁. * The second point is labeled (x₂, y₂) and is located at x = x₂ and y = y₂. A vertical dashed line extends from this point to the x-axis at x₂. * **Straight Line:** A thick blue straight line segment connects the point (x₁, y₁) to the point (x₂, y₂). This represents a secant line. * **Labels and Annotations:** * Δx is labeled along the horizontal distance between the vertical lines dropped from (x₁, y₁) and (x₂, y₂). * Δy is labeled along the vertical distance between the horizontal line through (x₁, y₁) and the point (x₂, y₂). * The formula m = Δy / Δx is labeled near the blue line segment. * The text "Not very good" is in a shaded rectangular box in the top left area of the chart. **Chart Description:** * **Type:** 2D Cartesian coordinate graph illustrating a function and concepts related to calculus (slope of a secant line and approaching a tangent line). * **Coordinate Axes:** * X-axis labeled 'x'. * Y-axis labeled 'y'. * Origin is at (0, 0). * X-axis ticks are marked at approximately 0, x₁, x₂, with some unmarked ticks between 0 and x₁, x₁ and x₂, and beyond x₂. * Y-axis ticks are marked at 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. * **Curve:** A black curved line represents the function f(x). It is concave up and passes through the first and second quadrants (though the visible portion is primarily in the first quadrant). * **Points:** * A white filled circle point on the curve labeled (x₁, y₁). * A white filled circle point on the curve at approximately (x₂, 4), labeled indirectly by lines extending to x₂ on the x-axis and a vertical line labeled Δy. * A third point is shown on the curve further to the right, approximately above x₃, labeled (x₃, y₃). A thick yellow arrow points towards this point along the tangent line originating from the point near (x₂, 4). * **Lines:** * A blue straight line connects the point (x₁, y₁) and the point near (x₂, 4). This represents a secant line. * A dashed horizontal line extends from the point (x₁, y₁) to the right, ending below the point near (x₂, 4). * A dashed vertical line extends upwards from x₂ on the x-axis to the point near (x₂, 4) on the curve. * A dashed horizontal line extends from the point near (x₁, y₁) to meet the dashed vertical line. * A thick yellow arrowed line originates near the point (x₂, 4) and extends upwards and to the right, approximately following the curve's direction at that point. It is labeled f'(x) and points towards the point (x₃, y₃) further along the curve. This represents the tangent line or derivative direction. * **Labels and Annotations:** * Curve labeled 'f(x)' near its upper right end. * Point labeled '(x₁, y₁)' below the leftmost white circle point. * Label 'Δx' below the dashed horizontal line between the vertical lines at x₁ and x₂. It represents the difference in x-coordinates (x₂ - x₁). * Label 'Δy' to the right of the dashed vertical line segment above the dashed horizontal line. It represents the difference in y-coordinates (y₂ - y₁). * Annotation bubble/box in the upper part of the graph contains the text "better ...". * Annotation text "move x₂ closer to x₁" below the "better ..." box and slightly above the curve, with an implied direction towards making the interval between x₁ and x₂ smaller. * Label 'f'(x)' near the thick yellow arrow. * Point labeled '(x₃, y₃)' near the tip of the thick yellow arrow on the curve. * **Relative Position and Direction:** * x₁ is to the left of x₂ on the x-axis. * The point (x₁, y₁) is to the lower left of the point near (x₂, 4) on the curve. * Δx is the horizontal distance between x₁ and x₂. * Δy is the vertical distance between the y-values at x₁ and x₂. * The blue secant line connects the two white circle points. * The yellow arrow points along the curve in the positive x and y direction, originating near the second point. * The text "move x₂ closer to x₁" suggests changing the position of x₂. **Chart Description:** * **Type:** Line graph representing a mathematical function. * **Coordinate Axes:** X-axis labeled 'x' and Y-axis labeled 'y'. Scales are indicated along both axes. * **Function:** A black curve representing a function is plotted, labeled 'f(x)'. The curve appears to be increasing and concave up. * **Points:** Two points are marked on the curve: * A point labeled (x₁, y₁) with coordinates (x₁, y₁), marked with a white circle. * A point labeled (x₂, y₂) with coordinates (x₂, y₂), marked with a white circle. * **Segments and Distances:** * A blue line segment connects the points (x₁, y₁) and (x₂, y₂). * The horizontal distance between x₁ and x₂ is labeled as Δx with a blue arrow below the segment. * The vertical distance between y₁ and y₂ is labeled as Δy with a blue arrow to the left of the segment. * **Annotations and Arrows:** * A gray text box at the top reads "... even better". * Text "move x₂ even closer to x₁" is present on the right side, accompanied by a thick yellow arrow pointing from the right (near x₂) towards the left (near x₁).