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![ Integration is an important concept
in mathematics and, together with its
inverse, differentiation, is one of the two main
operations in calculus.
Given a function f of a real variable x and an
interval [a, b] of the real line, the definite integral
is defined informally to be the area of the region in
the xy-plane bounded by the graph of f, the x-axis,
and the vertical lines x = a and x = b, such that areas
above the axis add to the total, and the area below
the x axis subtract from the total.](https://image.slidesharecdn.com/presentation1-120724115843-phpapp02/85/Isaac-newton-33-320.jpg)
More Related Content
Isaac newton
- 3. Sir Isaac Newton was an English physicist, mathematician, astronomer , natural philosopher, alchemist, and theologian, who has been considered by many to be the greatest and most influential scientist who ever lived. Newton described universal gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries.
- 4. Isaac Newton was born on what is retroactively considered 4 January 1643 at Woolsthorpe Manor in Woolsthorpe-by- Colsterworth, a hamlet in the county of Lincolnshire. At the time of Newton's birth, England had not adopted the Gregorian calendar and therefore his date of birth was recorded as Christmas Day, 25 December 1642. Newton was born three months after the death of his father, a prosperous farmer also named Isaac Newton.
- 5. Newton was educated at The King's School, Grantham , and in October 1659 he was removed from school. In June 1661, he was admitted to Trinity College, Cambridge as a sizar – a sort of work-study role. In 1665, he discovered the generalized binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague.
- 6. Newton received a bachelor’s degree at Trinity College, Cambridge in 1665 The next two years Newton returned home where he came up with most of his discoveries. He returned to Trinity College in 1667, where he became a professor of mathematics in 1669.
- 7. King school Trinity College
- 11. Calculus was invented by sir Isaac Newton Isaac Newton developed the use of calculus in his laws of motion and gravitation.
- 12. Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.
- 13. Calculus has historicallybeen called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus .
- 14. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
- 16. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are mere instructions, with no indication as to method, and some of them are wrong. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resembles the methods of integral calculus.
- 17. The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere.
- 19. In the 14th Century Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated many components of calculus such as the Taylor series, infinite series approximations, an integral test for convergence, early forms of differentiation, term by term integration, iterative methods for solutions of non- linear equations, and the theory that the area under a curve is its integral. Some consider the Yuktibhāṣā to be the first text on calculus.
- 20. Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.
- 21. Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on finite and infinitesimal analysis was written in 1748 by Maria Gaetana Agnesi
- 22. Sir Isaac Newton portrait
- 23. Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators.
- 25. The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
- 27. In geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point—that is, coincides with the curve at that point without crossing to the other side of the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point(c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
- 28. Tangent Tangent Tangent graph circle line
- 30. In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using divided differences. For any given set of data points, there is only one polynomial (of least possible degree) that passes through all of them. Thus, it is more appropriate to speak of "the Newton form of the interpolation polynomial" rather than of "the Newton interpolation polynomial". Like the Lagrange form, it is merely another way to write the same polynomial.
- 31. Newton's formula is of interest because it is the straightforward rate of change of its rate of change, etc. at one particular x value. Newton's formula is Taylor's polynomial based on finite differences instead of instantaneous rates of change.
- 33. Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that areas above the axis add to the total, and the area below the x axis subtract from the total.
- 35. A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan.
- 36. In the 1690s, Newton wrote a number of religious tracts dealing with the literal interpretation of the Bible. Henry Moore's belief in the Universe and rejection of Cartesian dualism may have influenced Newton's religious ideas. A manuscript he sent to John Locke in which he disputed the existence of the Trinity was never published. Later works – The Chronology of Ancient Kingdoms Amended (1728) and Observations Upon the Prophecies of Daniel and the Apocalypse of St. John (1733) – were published after his death. He also devoted a great deal of time to alchemy (see above). Newton was also a member of the Parliament of England from 1689 to 1690 and in 1701, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed.
- 38. Newton died in his sleep in London on 31 March 1727 and was buried in Westminster Abbey. Newton, a bachelor, had divested much of his estate to relatives during his last years, and died intestate. After his death, Newton's hair was examined and found to contain mercury, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life.
Editor's Notes
- #2 Animated floating petalsderGradient stops, click Add gradient stops or Remove gradient stops until two stops appear in the drop-down list.Also under Gradient stops, customize the gradient stops that you added as follows:Select the first stop in the slider, and then do the following:In the Position box, enter 0%.Click the button next to Color, and then under Theme Colors click Black, Text 1, Lighter 5% (sixth row, second option from the left).In the Transparency box, enter 40%.Select the last stop in the slider, and then do the following: In the Position box, enter 100%.Click the button next to Color, and then under Theme Colors click Black, Text 1, Lighter 5%(sixth row, second option from the left).In the Transparency box, enter 10%.Select the rectangle. On the Home tab, in the Drawing group, click Arrange, point to Align, and then do the following:Click Align Middle.Click Align Center. To reproduce the shape effects on this slide, do the following:On the Insert tab, in the Illustrations group, click Shapes, and then under Lines, click Freeform (11th option from the left).On the slide, click points to trace an outline around a shape on the background picture (in the example above, one of the flower petals). Right-click the freeform shape and click Edit Points. To make the freeform shape more similar to the shape on the picture, right-click a place to include another point, and click AddPoints. Drag the new point to follow the contour of the shape on the picture. To outline rounded edges of the shape on the picture, right-click a straight line segment, and click Curved Segment.Select the freeform shape. On the Home tab, in the bottom right corner of the Drawing group, click the Format Shape dialog box launcher. In the Format Shape dialog box, click Line Color in the left pane, and then in the Line Color pane, select No line. Also in the Format Shape dialog box, click Fill in the left pane, and then in the Fill pane, select Slide background fill. Repeat the process in steps 1-5to create six full-color shapes on the slide (in the example above, six flower petals from the same flower). To reproduce the animation effects on this slide, do the following:On the slide, select the rectangle. On the Animations tab, in the Advanced Animation group, click Add Animation, and then under Entrance click Fade.Also on theAnimations tab, in the Timing group, do the following:In theStartlist, select After Previous.In the Duration box, enter 2.00 seconds.On the slide, select one of the freeform shapes. On the Animations tab, in the Advanced Animation group, click Add Animation, and then under Emphasis click Spin.Also on the Animations tab, in the Timing group, do the following:In the Start list, select With Previous. In the Duration box, enter 2.00 seconds. On the Animations tab, in the Advanced Animation group, click Add Animation, and then under Emphasis click Grow/Shrink.Also on the Animations tab, in the Timing group, do the following:In the Start list, select With Previous. In the Duration box, enter 2.00 seconds. On the Animations tab, in the AdvancedAnimation group, click AddAnimation, and then click More Motion Paths. In the Add Motion Path dialog box, under Lines & Curves, click S Curve 1.Also on the Animations tab, in the Timing group, do the following:In the Start list, select With Previous. In the Duration box, enter 2.00 seconds. On the slide, select the S curve motion path. Point to the endpoint (red arrow) of the selected right motion path until the cursor becomes a two-headed arrow. Drag the endpoint off the right side of the slide.Right-click the motion path and select EditPoints, and then do the following:Right-click the middle point in the motion path animation and select SmoothPoint.Click the middle point. Drag one of the blue sizing lines until the curve in the motion path is smooth.On the slide, select the animated freeform shape. On the Animations tab, in the AdvancedAnimation group, click AnimationPainter, and then click one of the other freeform shapes. Repeat this process for each of the remaining freeform shapes.Also on the Animations tab, in the Advanced Animation group, click AnimationPane. In the Animation Pane, do the following to edit the animation effects:Select the second animation (first freeform shape spin effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Spin dialog box, do the following:On the Effect tab, in the Amount list, in the Custom box, enter 360⁰, and then press ENTER. Also in the Amount list, click Counterclockwise. On the Timing tab, in the Start list, select After Previous.Select the third animation (first freeform shape grow/shrink effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Grow/Shrink dialog box, do the following:On the Effect tab, in the Size list, in the Custom box, enter 40%, and then press ENTER. Also in the Size list, click Vertical. Select the fifth animation (second freeform shape spin effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Spin dialog box, do the following:On the Effect tab, in the Amount list, in the Custom box, enter 30⁰, and then press ENTER. Also in the Amount list, click Clockwise. On the Timing tab, in the Delay box, enter 0.1.Select the sixth animation (first freeform shape grow/shrink effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Grow/Shrink dialog box, do the following:On the Effect tab, in the Size list, in the Custom box, enter 40%, and then press ENTER. Also in the Size list, click Vertical. On the Timing tab, in the Delay box, enter 0.1.Select the seventh animation (first S-curve motion path effect). On the Animations tab, in the Timing group, in the Delay box, enter 0.1.Select the eighth animation (third freeform shape spin effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Spin dialog box, do the following:On the Effect tab, in the Amount list, in the Custom box, enter 40⁰, and then press ENTER. Also in the Amount list, click Clockwise. On the Timing tab, in the Delay box, enter 0.3.Select the ninth animation (third freeform shape grow/shrink effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Grow/Shrink dialog box, do the following:On the Effect tab, in the Size list, in the Custom box, enter 50%,and then press ENTER. Also in the Size list, click Horizontal. On the Timing tab, in the Delay box, enter 0.3.Select the 10th animation (first S-curve motion path effect). On the Animations tab, in the Timing group, in the Delay box, enter 0.3.Select the 11th animation (fourth freeform shape spin effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Spin dialog box, do the following:On the Effect tab, in the Amount list, in the Custom box, enter 90⁰, and then press ENTER. Also in the Amount list, click Counterclockwise. On the Timing tab, in the Delay box, enter 0.4.Select the 12th animation (fourth freeform shape grow/shrink effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Grow/Shrink dialog box, do the following:On the Effect tab, in the Size list, in the Custom box, enter 40%,and then press ENTER. Also in the Size list, click Vertical. On the Timing tab, in the Delay box, enter 0.4.Select the 13th animation (first S-curve motion path effect). On the Animations tab, in the Timing group, in the Delay box, enter 0.4.Select the 14th animation (fifth freeform shape spin effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Spin dialog box, do the following:On the Effect tab, in the Amount list, in the Custom box, enter 90⁰, and then press ENTER. Also in the Amount list, click Clockwise. On the Timing tab, in the Delay box, enter 0.5.Select the 15th animation (fifth freeform shape grow/shrink effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Grow/Shrink dialog box, do the following:On the Effect tab, in the Size list, in the Custom box, enter 50%,and then press ENTER. Also in the Size list, click Horizontal.On the Timing tab, in the Delay box, enter 0.5.Select the 16th animation (first S-curve motion path effect). On the Animations tab, in the Timing group, in the Delay box, enter 0.5.Select the 17th animation (sixth freeform shape spin effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Spin dialog box, do the following:On the Effect tab, in the Amount list, in the Custom box, enter 360⁰, and then press ENTER. Also in the Amount list, click Clockwise. On the Timing tab, in the Delay box, enter 0.6.Select the 18th animation (sixth freeform shape grow/shrink effect). On the Animations tab, in the Animation group, click the EffectOptions dialog box launcher. In the Grow/Shrink dialog box, do the following:On the Effect tab, in the Size list, in the Custom box, enter 50%,and then press ENTER. Also in the Size list, click Horizontal.On the Timing tab, in the Delay box, enter 0.6.Select the 19th animation (first S-curve motion path effect). On the Animations tab, in the Timing group, in the Delay box, enter 0.6.
- #3 Custom animation effects: duplicate picture with varied brightness to full color(Advanced)Tip: For the effects on this slide, use a picture with a “portrait” (vertical) orientation. To reproduce the text effects on this slide, do the following:On the Home tab, in the Slides group, click Layout, and then clickBlank. On the Insert tab, in the Text group, click Text Box, and then on the slide, drag to draw the text box.Enter text in the text box, and then select the text. On the Home tab, in the Font group, select Georgiafrom the Font list, enter 34 in the Font Size box, click Italic, click the arrow next to Font Color, and then under Theme Colors click Aqua, Accent 5, Darker 25% (fifth row, ninth option from the left).Drag the text box into the upper half of the slide, slightly left of the center.To reproduce the first picture with animation effects on this slide, do the following:On the Insert tab, in the Illustrations group, click Picture.In the Insert Picture dialog box, select a picture and then click Insert. On the slide, select the picture. Under Picture Tools, on the Format tab, in the bottom right corner of the Size group, click the Size and Position dialog box launcher. In the Size and Position dialog box, on the Size tab, resize or crop the picture as needed so that under Size and rotate, the Height box is set to 3.81” and the Width box is set to 2.54”. Resize the picture under Size and rotate by entering values into the Height and Width boxes. Crop the picture under Crop from by entering values into the Left, Right, Top, and Bottom boxes. Drag the picture into the into the top left corner of the slide. (Note: Align the left edge of the picture with the left edge of the slide, and leave a small margin between the top edge of the picture and the top edge of the slide.)Select the picture. Under PictureTools, on the Format tab, in the bottom right corner of the Picture Styles group, click the Format Shape dialog box launcher.In the Format Picture dialog box, in the left pane, click Picture, and then do the following in the Picture pane: Click the button next to Recolor, and then select the recolor option that works best with the picture. (Note: The first picture in the example above is set to No Recolor.)In the Brightness box, enter 50%. (Note: You may want to adjust the brightness settings for the picture you have chosen.)On the Animations tab, in the Animations group, click CustomAnimation.On the slide, select the picture, and then do the following in the Custom Animation task pane:Click Add Effect, point to Entrance, and then click MoreEffects. In the Add Entrance Effect dialog box,under Basic, click Fly In, and then click OK.Under Modify: Fly In, in the Start list, select With Previous. Under Modify: Fly In,in the Direction list, select From Left.Under Modify: Fly In,in the Speed list, select Fast.Click Add Effect, point to Entrance, and then click MoreEffects. In the Add Entrance Effect dialog box,under Subtle, click Expand, and then click OK.Under Modify: Expand, in the Start list, select With Previous.Under Modify: Expand, in the Speed list, select Fast.To reproduce the second picture with animation effects on this slide, do the following:On the slide, select the picture. On the Home tab, in the Clipboard group, click the arrow under Paste, and then click Duplicate.Drag the second picture until the left side slightly overlaps the right side of the first picture.Select the second picture. Under PictureTools, on the Format tab, in the bottom right corner of the Picture Styles group, click the Format Shape dialog box launcher.In the Format Picture dialog box, in the left pane, click Picture. In the Picture pane, in the Brightness box, enter 40%. (Note: You may want to adjust the brightness settings for the picture you have chosen.)Press and hold CTRL, and then select the third and fourth animation effects (fly-in and expand effects for the second picture) in the CustomAnimation task pane. Click the arrow to the right of one of the animation effects, and then click Timing. In the Effect Options dialog box, on the Timing tab, in the Delay box, enter 0.2, and then click OK.To reproduce the third picture with animation effects on this slide, do the following:On the slide, select the second picture. On the Home tab, in the Clipboard group, click the arrow under Paste, and then click Duplicate.Drag the third picture until its left side slightly overlaps the right side of the second picture.Select the third picture. Under PictureTools, on the Format tab, in the bottom right corner of the Picture Styles group, click the Format Shape dialog box launcher.In the Format Picture dialog box, in the left pane, click Picture. In the Picture pane, in the Brightness box, enter 20%. (Note: You may want to adjust the brightness settings for the picture you have chosen.)Press and hold CTRL, and then select the fifth and sixth animation effects (fly-in and expand effects for the third picture) in the CustomAnimation task pane. Click the arrow to the right of one of the animation effects, and then click Timing. In the Effect Options dialog box, on the Timing tab, in the Delay box, enter 0.4, and then click OK.To reproduce the fourth picture with animation effects on this slide, do the following:On the slide, select the third picture. On the Home tab, in the Clipboard group, click the arrow under Paste, and then click Duplicate.Drag the fourth picture until its left side slightly overlaps the right side of the third picture.Select the fourth picture. Under PictureTools, on the Format tab, in the bottom right corner of the Picture Styles group, click the Format Shape dialog box launcher.In the Format Picture dialog box, in the left pane, click Picture. In the Picture pane, in the Brightness box, enter 0%. Press and hold CTRL, and then select the seventh and eighth animation effects (fly-in and expand effects for the fourth picture) in the CustomAnimation pane. Click the arrow to the right of one of the animation effects, and then click Timing. In the Effect Options dialog box, on the Timing tab, in the Delay box, enter 0.6, and then click OK.To reproduce the fifth picture with animation effects on this slide, do the following:On the slide, select the fourth picture. On the Home tab, in the Clipboard group, click the arrow under Paste, and then click Duplicate.Drag the fifth picture until its left side slightly overlaps the right side of the fourth picture.Select the fifth picture. Under PictureTools, on the Format tab, in the bottom right corner of the Picture Styles group, click the Format Shape dialog box launcher.In the Format Picture dialog box, in the left pane, click Picture. In the Picture pane, click the button next to Recolor, and then under No Recolor click No Recolor. Press and hold CTRL, and then select the ninth and 10th animation effects (fly-in and expand effects for the fifth picture) in the CustomAnimation task pane. Click the arrow to the right of one of the selected animation effects, and then click Timing. In the Effect Options dialog box, on the Timing tab, in the Delay box, enter 0.8, and then click OK.To reproduce the animation effects for the text box on this slide, do the following:On the Home tab, in the Editing group, click Select, and then click Selection Pane.In the Selection and Visibility pane, select the text box, and then do the following in the CustomAnimation task pane:Click Add Effect, point to Entrance, and then click MoreEffects. In the Add Entrance Effect dialog box,under Subtle, click Fade, and then click OK.Under Modify: Fade, in the Start list, select With Previous. Under Modify: Fade, in the Speed list, select Very Fast.In the Selection and Visibility pane, press and hold CTRL, and then select the first, second, third, and fourth pictures. In the CustomAnimation task pane do the following:Click Add Effect, point to Exit, and then click MoreEffects. In the Add Exit Effect dialog box,under Moderate, click Stretchy, and then click OK.Under Modify: Stretchy, in the Start list, select With Previous. Under Modify: Stretchy, in the Speed list, select Fast.Also in the Custom Animation task pane, select the 12th animation effect (first stretchy effect for the text box). Click the arrow to the right of the animation effect, and then click Timing. In the Stretchy dialog box, on the Timing tab, in the Delay box, enter 1.4, and then click OK.Also in the Custom Animation task pane, select the 13th animation effect (second stretchy effect for the text box). Click the arrow to the right of the animation effect, and then click Timing. In the Stretchy dialog box, on the Timing tab, in the Delay box, enter 1.6, and then click OK.Also in the Custom Animation task pane, select the 14th animation effect (third stretchy effect for the text box). Click the arrow to the right of the animation effect, and then click Timing. In the Stretchy dialog box, on the Timing tab, in the Delay box, enter 1.8, and then click OK.Also in the Custom Animation task pane, select the 15th animation effect (fourth stretchy effect for the text box). Click the arrow to the right of the animation effect, and then click Timing. In the Stretchy dialog box, on the Timing tab, in the Delay box, enter 2.0, and then click OK.To reproduce the background effects on this slide, do the following:Right-click the slide background area, and then click Format Background. In the Format Background dialog box, click Fill in the left pane, select Gradient fill in the right pane, and then do the following:In the Type list, select Radial.Click the button next to Direction, and then click From Center (third option from the left).Under Gradient stops, click Add or Remove until three stops appear in the drop-down list.Also under Gradient stops, customize the gradient stops as follows:Select Stop 1 from the list, and then do the following:In the Stop position box, enter 0%.Click the button next to Color, and then under Theme Colors click White, Background 1, Darker 5%(second row, first option from the left). Select Stop 2 from the list, and then do the following: In the Stop position box, enter 34%.Click the button next to Color, and then under Theme Colors click White, Background 1, Darker 15% (third row, first option from the left). Select Stop 3 from the list, and then do the following: In the Stop position box, enter 100%.Click the button next to Color, and then under Theme Colors click White, Background 1, Darker 50% (sixth row, first option from the left).