Differential calculus. A byproduct of these extraction methods was that large amounts of gravel, silt, heavy metals, and other pollutants went into streams and rivers. , many areas still bear the scars of hydraulic mining, since the resulting exposed earth and downstream gravel deposits do not support plant life. It is estimated that by 1855, at least 300,000 gold-seekers, merchants, and other immigrants had arrived in California from around the world. The huge numbers of newcomers brought by the Gold Rush drove Native Americans out of their traditional hunting, fishing and food-gathering areas. To protect their homes and livelihood, some Native 	Sierra Nevada
8943189	Americans responded by attacking the miners, provoking counter-attacks on native villages. The Native Americans, out-gunned, were often slaughtered. Thorough exploration. The Gold Rush populated the western foothills of the Sierra Nevada, but even by 1860, most of the Sierra was unexplored. The state legislature authorized the California Geological Survey to officially explore the Sierra (and survey the rest of the state). Josiah Whitney was appointed to head the survey. Men of the survey, including William H. Brewer, Charles F. Hoffmann and Clarence King, explored the backcountry of what would become Yosemite National Park in 1863. In 1864, they 	Sierra Nevada
8943190	explored the area around Kings Canyon. In 1869, John Muir started his wanderings in the Sierra Nevada range, and in 1871, King was the first to climb Mount Langley, mistakenly believing he had summited Mount Whitney, the highest peak in the range. In 1873, Mount Whitney was climbed for the first time by 3 men from Lone Pine, CA on a fishing trip. From 1892 to 1897 Theodore Solomons made the first attempt to map a route along the crest of the Sierra. Other people finished exploring and mapping the Sierra. Bolton Coit Brown explored the Kings River watershed in 	Sierra Nevada
8943191	1895–1899. Joseph N. LeConte mapped the area around Yosemite National Park and what would become Kings Canyon National Park. James S. Hutchinson, a noted mountaineer, climbed the Palisades (1904) and Mount Humphreys (1905). By 1912, the USGS published a set of maps of the Sierra Nevada, and the era of exploration was over. Conservation. The tourism potential of the Sierra Nevada was recognized early in the European history of the range. Yosemite Valley was first protected by the federal government in 1864. The Valley and Mariposa Grove were ceded to California in 1866 and turned into a state park. John 	Sierra Nevada
8943192	Muir perceived overgrazing by sheep and logging of giant sequoia to be a problem in the Sierra. Muir successfully lobbied for the protection of the rest of Yosemite National Park: Congress created an Act to protect the park in 1890. The Valley and Mariposa Grove were added to the Park in 1906. In the same year, Sequoia National Park was formed to protect the Giant Sequoia: all logging of the Sequoia ceased at that time. In 1903, the city of San Francisco proposed building a hydroelectric dam to flood Hetch Hetchy Valley. The city and the Sierra Club argued over 	Sierra Nevada
8943193	the dam for 10 years, until the U.S. Congress passed the Raker Act in 1913 and allowed dam building to proceed. O'Shaughnessy Dam was completed in 1923. Between 1912 and 1918, Congress debated three times to protect Lake Tahoe in a national park. None of these efforts succeeded, and after World War II, towns such as South Lake Tahoe grew around the shores of the lake. By 1980, the permanent population of the Lake Tahoe area grew to 50,000, while the summer population grew to 90,000. The development around Lake Tahoe affected the clarity of the lake water. In order 	Sierra Nevada
8943194	to preserve the lake's clarity, construction in the Tahoe basin is currently regulated by the Tahoe Regional Planning Agency. As the 20th century progressed, more of the Sierra became available for recreation; other forms of economic activity decreased. The John Muir Trail, a trail that followed the Sierra crest from Yosemite Valley to Mount Whitney, was funded in 1915 and finished in 1938. Kings Canyon National Park was formed in 1940 to protect the deep canyon of the Kings River. In the 1920s, automobile clubs and nearby towns started to lobby for trans-Sierra highways over Piute Pass (which 	Sierra Nevada
8943195	would have closed the gap in SR 168) and other locations. However, by end of the 1920s, the Forest Service and the Sierra Club decided that roadless wilderness in the Sierra was valuable, and fought the proposal. The Piute Pass proposal faded out by the early 1930s, with the Forest Service proposing a route over Minaret Summit in 1933. The Minaret Summit route was lobbied against by California's Governor Ronald Reagan in 1972. The expansion of the John Muir and Ansel Adams Wildernesses in the 1980s sealed off the Minaret Summit route. A trans-Sierra route between Porterville and 	Sierra Nevada
8943196	Lone Pine was proposed by local businessmen in 1923. Eventually, a circuitous route across the Sierra was built across Sherman Pass by 1976. By 1964, the Wilderness Act protected portions of the Sierra as primitive areas where humans are simply temporary visitors. Gradually, 20 wilderness areas were established to protect scenic backcountry of the Sierra. These wilderness areas include the John Muir Wilderness (protecting the eastern slope of the Sierra and the area between Yosemite and Kings Canyon Parks), and wilderness within each of the National Parks. The Sierra Nevada still faces a number of issues that threaten its conservation	Sierra Nevada
8943197	. Logging occurs on both private and public lands, including controversial clearcut methods and thinning logging on private and public lands. Grazing occurs on private lands as well as on National Forest lands, which include Wilderness areas. Overgrazing can alter hydrologic processes and vegetation composition, remove vegetation that serves as food and habitat for native species, and contribute to sedimentation and pollution in waterways. A recent increase in large wildfires like the Rim Fire in Yosemite National Park and the Stanislaus National Forest and the King Fire on the Eldorado National Forest, has prompted concerns. A 2015 study indicated that the 	Sierra Nevada
8943198	increase in fire risk in California may be attributable to human-induced climate change. A study looking back over 8,000 years found that warmer climate periods experienced severe droughts and more stand-replacing fires and concluded that as climate is such a powerful influence on wildfires, trying to recreate presettlement forest structure is likely impossible in a warmer future. Sierra Nevada The Sierra Nevada () is a mountain range in the Western United States, between the Central Valley of California and the Great Basin. The vast majority of the range lies in the state of California, although the Carson Range spur 	Sierra Nevada
8943199	Differential calculus Differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at 	Differential calculus
8943200	a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration. Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving 	Differential calculus
8943201	body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives 	Differential calculus
8943202	are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra. Derivative. The derivative of formula_1 at the point formula_2 is the slope of the tangent to formula_3. In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form formula_4. The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in formula_5 	Differential calculus
8943203	by the change in formula_6, meaning that formula_7. For, the graph of formula_8 has a slope of formula_9, as shown in the diagram below: For brevity, formula_11 is often written as formula_12, with formula_13 being the Greek letter Delta, meaning 'change in'. The slope of a linear equation is constant, meaning that the steepness is the same everywhere. However, many graphs, for instance formula_14, vary in their steepness. This means that you can no longer pick any two arbitrary points and compute the slope. Instead, the slope of the graph can be computed by considering the tangent line—a line 	Differential calculus
8943204	that 'just touches' a particular point. The slope of a curve at a particular point is equal to the slope of the tangent to that point. For example, formula_14 has a slope of formula_16 at formula_17 because the slope of the tangent line to that point is equal to formula_16: The derivative of a function is then simply the slope of this tangent line. Even though the tangent line only touches a single point at the point of tangency, it can be approximated by a line that goes through two points. This is known as a secant line. If the 	Differential calculus
8943205	two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar: The advantage of using a secant line is that its slope can be calculated directly. Consider the two points on the graph formula_19 and formula_20, where formula_21 is a small number. As before, the slope of the line passing through these two points can be calculated with the formula formula_22. This gives As formula_21 gets closer and closer to formula_25, the slope of the secant line gets closer and closer 	Differential calculus
8943206	to the slope of the tangent line. This is formally written as The above expression means 'as formula_21 gets closer and closer to 0, the slope of the secant line gets closer and closer to a certain value'. The value that is being approached is the derivative of formula_1; this can be written as formula_29. If formula_30, the derivative can also be written as formula_31, with formula_32 representing an infinitesimal change. For example, formula_33 represents an infinitesimal change in x. In summary, if formula_30, then the derivative of formula_1 is provided such a limit exists. We have thus succeeded in 	Differential calculus
8943207	properly defining the derivative of a function, meaning that the 'slope of the tangent line' now has a precise mathematical meaning. Differentiating a function using the above definition is known as differentiation from first principles. Here is a proof, using differentiation from first principles, that the derivative of formula_14 is formula_38: As formula_21 approaches formula_25, formula_42 approaches formula_38. Therefore, formula_44. This proof can be generalised to show that formula_45 if formula_46 and formula_47 are constants. This is known as the power rule. For example, formula_48. However, many other functions cannot be differentiated as easily as polynomial functions, meaning that 	Differential calculus
8943208	sometimes further techniques are needed to find the derivative of a function. These techniques include the chain rule, product rule, and quotient rule. Other functions cannot be differentiated at all, giving rise to the concept of differentiability. A closely related concept to the derivative of a function is its differential. When and are real variables, the derivative of at is the slope of the tangent line to the graph of at . Because the source and target of are one-dimensional, the derivative of is a real number. If and are vectors, then the best linear approximation to the graph 	Differential calculus
8943209	of depends on how changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted . The linearization of in all directions at once is called the total derivative. History of differentiation. The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287–212 BC) and Apollonius of Perga (c. 262–190 BC). Archimedes also made use of indivisibles, although these were primarily used to study areas and volumes rather than derivatives 	Differential calculus
8943210	and tangents (see The Method of Mechanical Theorems"). The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 AD, when the astronomer and mathematician Aryabhata (476–550) used infinitesimals to study the orbit of the Moon. The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114–1185); indeed, it has been argued that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem". The mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), in his "Treatise on Equations", established  conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He obtained, for example, that the maximum (for positive ) of the cubic occurs when , and concluded therefrom that the equation has exactly one positive solution when , and two positive solutions whenever . The historian of science, Roshdi Rashed, has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the  function to be known. The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716), who provided independent and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Ibn al-Haytham (Alhazen). For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Pierre de Fermat (1607-1665), Isaac Barrow  (1630–1677), René Descartes (1596–1650), Christiaan Huygens (1629–1695), Blaise Pascal (1623–1662) and John Wallis (1616–1703). Regarding Fermat's influence, Newton once wrote in a letter that ""I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general."" Isaac Barrow is generally given credit for the early development of the derivative. Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the  notation still used today. Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789–1857), Bernhard Riemann (1826–1866), and Karl Weierstrass (1815–1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane. Applications of derivatives. Optimization. If is a differentiable function on (or an open interval) and is a local maximum or a local minimum of , then the derivative of at is zero. Points where are called "critical points  or stationary points" (and the value of at is called a "critical value"). If is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points. If is twice differentiable, then conversely, a critical point of can be analysed by considering the second derivative of at : This is called the second derivative test. An alternative approach, called the first derivative test, involves considering the sign of the on each side of the critical point. Taking derivatives and solving for critical points is therefore often a simple way to find local minima  or maxima, which can be useful in optimization. By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints. This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points. In higher dimensions, a critical point of a scalar valued  function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a "saddle point", and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is  considered to be inconclusive. Calculus of variations. One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear. These paths are called geodesics, and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This  surface is called a minimal surface and it, too, can be found using the calculus of variations. Physics. Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the "time derivative" — the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics: For example, if an object's position on a line is given  by then the object's velocity is and the object's acceleration is which is constant. Differential equations. A differential equation is a relation between a collection of functions and their derivatives. An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton's second law, which describes the relationship between  acceleration and force, can be stated as the ordinary differential equation The heat equation in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation Here is the temperature of the rod at position and time and is a constant that depends on how fast heat diffuses through the rod. Mean value theorem. The mean value theorem gives a relationship between values of the derivative and values of the original function. If is a real-valued function and and are numbers with , then the mean value theorem says that under mild hypotheses, the  slope between the two points and is equal to the slope of the tangent line to at some point between and . In other words, In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose that has derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph of must equal the slope of one of the tangent  lines of . All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function. Taylor polynomials and Taylor series. The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is  to take a quadratic approximation. That is to say, the linearization of a real-valued function at the point is a linear polynomial , and it may be possible to get a better approximation by considering a quadratic polynomial . Still better might be a cubic polynomial , and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients , , , and that makes the approximation as good as possible. In the neighbourhood of , for the best possible choice is always , and for the best possible choice is always . For  , , and higher-degree coefficients, these coefficients are determined by higher derivatives of . should always be , and should always be . Using these coefficients gives the Taylor polynomial of . The Taylor polynomial of degree is the polynomial of degree which best approximates , and its coefficients can be found by a generalization of the above formulas. Taylor's theorem gives a precise bound on how good the approximation is. If is a polynomial of degree less than or equal to , then the Taylor polynomial of degree equals . The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor  series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic functions. It is impossible for functions with discontinuities or sharp corners to be analytic; moreover, there exist smooth functions which are also not analytic. Implicit function theorem. Some natural geometric shapes, such as circles, cannot be drawn as the graph of a function. For instance, if , then the circle is the set of all pairs such that . This set is called the zero set of , and is not the same as the graph of , which is a  paraboloid. The implicit function theorem converts relations such as into functions. It states that if is continuously differentiable, then around most points, the zero set of looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of . The circle, for instance, can be pasted together from the graphs of the two functions . In a neighborhood of every point on the circle except and , one of these two functions has a graph that looks like the circle. (These two functions also happen to meet and , but this is not  guaranteed by the implicit function theorem.) The implicit function theorem is closely related to the inverse function theorem, which states when a function looks like graphs of invertible functions pasted together. Differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a