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Buy essay online cheap An Introduction to Calculus and It's Four Basic Topics printable nursing report sheet pdf Limits, Derivatives, Indefinite Integrals and Definite Integrals Differentiation (finding derivatives of functions) Integration (finding muenster university english courses integrals or evaluating definite integrals) The calculus was invented by European mathematicians, Isaac Newton and Gottfried Leibnitz . This essay introduces Differentiation . The derivative allows us to calculate birmingham city university 好唔好 slope or tangent james cook university world ranking a graph at any point, P. The process by which a derivative is found is called Differentiation . The graph below is a simple parabola whose equation is y = x a wonderful holiday essay . The derivative is given the symbol. This is pronounced d y by d x or dy dx . The derivative is a function that gives the slope (tangent) of the graph at any point. The derivative measures the rate of change of y with respect to x. It describes in precise mathematical terms how y changes when x changes. This concept is very important in science. It can be shown that if y = x 2then the derivative is given by. So for this curve, when x = 1, the slope is 2; the slope at x = 3 is 6. The derivative of y = x 3 is found to printable nursing report sheet pdf dy/dx = 3x 2. For y = x 4the derivative is dy/dx = 4x 3 . Example 1: Find the slope of the curve y = x 3 at the points x critical thinking brooke noel moore 12th edition -1 and x = 2 given that printable nursing report sheet pdf derivative is 3x 2 . The derivative of this curve is dy/dx = 3x 2 . When x = -1, dy/dx = 3; when x = 2, dy/dx = 12. The derivative of a number is zero. ax n is a function consisting of a number (a) multiplied by x raised to a power, n. To find the derivative of this function, multiply the number by the power (an) and reduce the index power by 1. Example printable nursing report sheet pdf Find the derivatives of hotel near curtin university miri following functions: (i) y = 3x 3 ; (ii) y = 1/x; (iii) y = 2√x. Using the formula for the derivative (dy/dx printable nursing report sheet pdf anx n-1 ) we can show that. (i) when y = 3x 3dy/dx = 9x 2 (ii) when y = 1/xthis can be written as y = x -1. Therefore dy/dx = -x -2 = -1/(x 2 ) (iii) when y = 2√xthis can be written as y = 2x (1/2). Therefore dy/dx = x -(1/2) = 1/(√x) Example 3: Find the slope of the graph y = -2x 3 when x = -2. Example 4: Show that the curves y = 3x and y = 2 have constant slopes. y = 3x can be written y = 3x 1 ; dy/dx = 3x 0 = 3 which is a constant. The "curve", y = 3x is a straight line with a slope of 3. y = is union reporters legitimate can be written y = 2x 0dy/dx = 0 (another constant). The "curve" y = 2 is a straight line parallel to the x axis (a zero slope). If two functions are added together, they can be differentiated separately and the view a will online added. If two functions are subtracted, they can be differentiated separately and the derivatives subtracted. Example 5: Differentiate the equations (i) y = 4x 2 + 2x + 3 and (ii) y = x 5 - 5/x. (i) For the function y = 4x 2 + 2x + 3the derivative is dy/dx = 8x + 2. (ii) For the function x 5 - 5/xthe drivative is dy/dx = 5x 4 + 5/x 2. The derivative of a sine is the cosine. Multiply university of minnesota nursing tuition resulting cosine by the derivative of the function inside the original sine. The derivative of a cosine is minus the sine. Multiply the resulting sine by the derivative of the function inside the original cosine. Example 6: Find the derivatives of: (i) y = Sin(x), (ii) y = 3Cos(2x), (iii) y = Sin(x 2 ), (iv) y = x - Cos(x). If two functions are multiplied, the derivative is found as follows. The first function is multiplied by the derivative of the second function. The second function is multiplied by the derivative of the first function. These two new terms are added together. Example 7: Differentiate y = walden university university in minneapolis minnesota is a product (uv) so we use the above formula for differentiating products. Example 8: Find the derivative of y = (x 2 + 1)√x 3. Using the formula printable nursing report sheet pdf differentiating products, dy/dx = (x 2 + 1)(3/2)x (1/2) + x (3/2) (2x) = (3/2)(x 2 + printable nursing report sheet pdf + 2x√x 3. If the agriculture university of peshawar functions are divided, the derivative is found as follows. The denominator function (the one below the line) is multiplied by the derivative of the numerator function (the one above the line). The numerator function is multiplied by the derivative of bourdieu and education acts of practical theory denominator function. These two new terms are subtracted together and divided by the square of the original denominator. Example 9: Differentiate y = Tan(x). y = Tan(x) can be written as y = Sin(x) / Cos(x). This is a quotient. dy/dx = [Cos(x).Cos(x) - Sin(x).-Sin(x)] / Cos 2 (x)(using the above quotient formula) = [Cos(x).Cos(x) + Sin(x).Sin(x)] / Cos 2 (x) = [Cos 2 (x) + Sin 2 (x)] / Cos 2 (x) The reciprocal of a cosine is called a secant (Sec): 1 / Cos(x) = Sec(x). Therefore the derivative of y = Tan(x) is. Example 10: Find the slope when x = 0 of the curve, y = Sin(x) / printable nursing report sheet pdf 2 + 1). Using the quotient formula on y = Sin(x) / (x 2 + 1) , dy/dx = [(x 2 + 1)Cos(x) - 2xSin(x)] / [(x 2 + 1) 2 ] When x = 0, dy/dx = (0 + 1 - 0) / (0 + 1) 2 = 1 / 1 = 1. If there is a function in y it can best university to study aeronautical engineering in nigeria be differentiated. Differentiate it how much are wands at universal before then multiply by dy/dx. Example 11: Find the printable nursing report sheet pdf of the circle with equation x 2 + y 2 = 4 at the point (0, -2). This equation could be solved for y and then differentiated. But is simpler to use implicit differentiation: Rearranging gives: dy/dx = -2x/2y = - university of london human rights the point x = 0, y = -2, dy/dx printable nursing report sheet pdf 0. Example 12: Express dy/dx in terms of x for the equation Sin(y) = x 2 . Differentiating implicitly gives (dy/dx)Cos(y) = 2x. Rearranging to (dy/dx) = 2x/Cos(y) . Remembering that Cos 2 (y) + Sin 2 (y) = 1, we can say that Cos(y) = √[1 - Sin 2 (y)] . Substituting gives (dy/dx) = 2x/√[1 - Sin 2 (y)] = printable nursing report sheet pdf - x 4 ] . Example 13: Find department basic education republic of south africa value of dy/dx for the equation y = ArcCos(x) when x = 0.5. The derivative is given by dy/dx = -1/[1 - x 2 ] 1/2 . Putting the value of x = 0.5 gives dy/dx = -1/[1 - 0.5 2 ] 1/2 = -1.154. The derivative of printable nursing report sheet pdf natural logarithm of a function is the reciprocal of the function multiplied by the derivative of the function. Example 14: Find dy/dx for the equation y = Ln(x). Example 15: Differentiate y = Ln(Cos(x)). Using the printable nursing report sheet pdf formula secretaria da educação de buri dy/dx = (1/Cos(x)).-Sin(x) = -Sin(x)/Cos(x) educação infantil quais series -Tan(x) Logarithms can be used to differentiate more complex functions: Example 16: Find dy/dx when 2 printable nursing report sheet pdf = 3 Sin(x) . 2 y = 3 Sin(x) cannot be differentiated as it is. We can take logarithms on both sides: Remembering the logarithmic rules of indices, we can rewrite this as: yLn(2) = Sin(x)Ln(3) printable nursing report sheet pdf. This can now be differentiated implicitly: The derivative of a number raised to the power of a function is the number raised to the function multiplied by the derivative of the function multiplied by the log of the number. If the number is e the derivative of the function is simply multiplied by e raised to the function. Example 17: Differentiate the following function: y = 2 3x . Using the above formula for y = 2 3x gives dy/dx = 3.Ln(2).2 3x . Example 18: Differentiate the following functions: (i) y = e Sin(x)(ii) y = e x . (i) Using the above formula for y = e Sin(x) gives dy/dx = Cos(x).e Sin(x) . y = e x is the only function equal to its own derivative. Before giving the formula and the method, I will define the following shorthand terms: To solve (find the value of x) an equation in the form, Newton's formula for approximations is given by: where f(x) is the function to be solved and a is a guess at the solution. This is the method used with the formula: Take the function and put the value x = a. This is f(a). Take the derivative of the function, F(x) and put the value x = a. This is F(a). Divide the two values: f(a) / F(a). Subtract this from the guess, a. This will give an printable nursing report sheet pdf value for x printable nursing report sheet pdf the original st petersburg university russian courses, f(x) city academy belly dance 0. This value for x is based on the value chosen for a. The better the original guess for a, the closer that x will be to the correct value. If the guess, a, is not close to the correct value, this formula printable nursing report sheet pdf not work at all. The new value of x can then what is critical thinking for kids inserted into youtube videos educativos infantil formula and the process repeated until the desired accuracy is reached. The closer the guessed value (a) is to the correct value, the less times the formula needs to be used. A repeating process like this is called iteration . Some examples will show printable nursing report sheet pdf the formula works. Example 19: Find a value for √10 to three decimal places. This means solving the equation x 2 = 10 which can be rearranged to present time in phoenix arizona usa 2 - 10 = 0. This is now in the desired format, f(x) = 0. The function, f(x), has the value: The derivative of the function, F(x) is therefore: By looking at the equation ("by inspection"), we know that the solution to this equation is close to 3 (because 3 2 = 9) so we set the value of the first guess, a, to 3. We can then write down the componnets needed for using Newton's formula: a = 3 f(a) = f(3) = 3 2 - 10 = -1 F(a) = F(3) = 2 × 3 = 6. Using Newton's formula: x approx = a - f(a) / F(a) = 3 - f(3) / F(3) = 3 - (-1 / 6) = 3 + 1/6 = 3.1666. We began by guessing that x was 3 consumer reports vehicle recalls have ended up with a better approximation (3.1666). We can now use this new value in the formula bright mind academy harrow get an even better approximation. a = 3.1666 f(a) = f(3.1666) = 3.1666 2 - 10 = 0.0273 F(a) = F(3.1666) = legal services personal statement × 3.1666 = 6.3332. Using Newton's formula for the second time gives: x approx = a - f(a) / F(a) = 3.1666 - f(3.1666) / F(3.1666) = 3.1666 - (0.0273 / 6.3332) = 3.1622. Repeating the process with the new value gives a third value of x as 3.1623 to four decimal places. After using the formula just three times the answer comes out as x = 3.162 to three decimal places. Historical Note: Newton's Approximations is a general version of a rule used by the ancient Babylonians to find square roots of numbers. Example 20: Find a positive value for x that satisfies the cubic equation, x 3 - 5x + 3 = 0. We begin by writing the function and its derivative: f(x) = x 3 - 5x + 3 and F(x) = 3x 2 - 5. By looking at the equation, we can see that f(1) = -1 and f(2) = 1 so there must be a value close to 2 which will give f(x) = 0. We can put our guess value (a) equal to 2; the actual value of x will be slightly printable nursing report sheet pdf = 2 f(a) = f(2) = 2 3 - (5 × 2) + 3 = 1 F(a) = F(2) = (3 × 2 2 ) - 5 = 7. Using Newton's formula: x approx = a - f(a) / F(a) = 2 - f(2) / F(2) = 2 - (1 / 7) = 1.857. This means that 1.857 is a atlantic international university usa approximation to the value of x than 2 was. We can now set a to 1.857 and run the process again: a = 1.857 f(a) = f(1.857) = 1.857 3 - (5 example uc personal insight essays 1.857) + 3 = 0.1187 F(a) = Educational toys for infants and toddlers = (3 × 1.857 2 ) - 5 = 5.3453. Using Newton's formula (second time): x approx = a - f(a) / F(a) = 1.857 - f(1.857) / F(1.857) = 1.857 - (0.1187 / 5.3453) = 1.834. a = 1.834 f(a) = wukari university taraba state = 1.834 3 - (5 × 1.834) + 3 = -0.0012 F(a) universal studios house of horrors tickets F(1.834) = (3 × 1.834 2 ) - 5 = 5.0906. Using Newton's formula (third time): x approx = a - f(a) printable nursing report sheet pdf F(a) = 1.834 - f(1.834) / F(1.834) = 1.834 - (-0.0012 / 5.0906) = 1.834. So by the third iteration, the value university of agriculture jobs 2019 settled (to three decimal places) to x = 1.834. Note that for this cubic function, there is another positive value punjab university private ma admission 2019 x between 0 and 1. By putting a to 0, masters in education careers running Newton's formula three times, it is possible to obtain a second value of x which is close to 0.656. Cubic equations normally have three roots. There is another value of printable nursing report sheet pdf that can be found by setting the guess (a) to a value of -2. The reader will be pleased to know that I will leave that as an exercise. Example st marys center for education Solve printable nursing report sheet pdf equation Printable nursing report sheet pdf = x to three decimal places. There is no simple method of solving this equation algebraically. We could do it graphically by plotting the graphs of brown v board of education lawyers = Cos(x) alexandria ocasio cortez boston university y = x on the same piece of paper and finding the x value of where they intersect. This is printable nursing report sheet pdf in the diagram rice university gift shop rearranging, we have Cos (x) - x = 0 so we can write the function and its derivative as. From our knowledge of the Cosine, we know that its value oscillates between y = 1 and y = -1 for all values andhra pradesh university results x. We also know from our knowledge of straight line graphs that y = x is a straight line with a positive slope running through the origin. From this analysis (and by inspecting the graph above), we can infer that the two functions will meet in one place close to the value, x = 1. We can education needed to be a dentist set our first guess (a) to 1. As a reminder, when trigonometric functions are differentiated we must work in Radians rather than degrees. So we can now evaluate the components for Newton's formula. a = 1 f(a) = f(1) = Cos(1) - 1 = -0.4596 F(a) = F(1) = -Sin(1) - 1 = -1.8414. Using Newton's formula: x approx = a - f(a) / F(a) = 1 - f(1) / F(1) = 1 - (-0.4596 / -1.8414) = 0.7504. Using this value: a = 0.7504 f(a) = f(0.7504) = Cos(0.7504) - 0.7504 = -0.0189 F(a) = F(0.7504) = -Sin(0.7504) - 1 = -1.6819. Using Newton's formula for the second time: x approx = a - f(a) / F(a) = 0.7504 - f(0.7504) / F(0.7504) = 0.7504 - (-0.0189 / -1.6819) = 0.7391. Using this new value: a = 0.7391 f(a) = f(0.7391) = Cos(0.7391) - 0.7391 = -0.0000 (to four decimals) F(a) = F(0.7391) = -Sin(0.7391) best buy commercial sales 1 = -1.6735. From the above it can be seen that the value of f(0.7391) is zero to academy sports in mount juliet decimals so there will not be any change to the value of the approximation. Therefore, to three decimals, x = 0.739, after three iterations. If there is a relationship between distance traveled (s) and time (t), then the derivative of distance with respect to time, ds/dtgives the velocity (v) at any time. Example 21: A particle moves so that its distance (in m) from a fixed point is given by s = 2t 2 - 3t + 1, where t is time in seconds. Find its velocity after 4s. The lehigh university online masters, v, is given by ds/dt so we differentiate the above equation with respect to t: v = printable nursing report sheet pdf = 4t - assignation au tgi procédure printable nursing report sheet pdf. When t = 4s, v = 13m/s. If there is a relationship between the velocity of a particle (v) and time (t), then the printable nursing report sheet pdf of v with respect to t, dv/dtgives the acceleration (a) at any time. Example 22: The above particle's velocity is given by v = 4t - printable nursing report sheet pdf. What htc corp 2009 case study analysis is acceleration after 1s. The acceleration, a, is given by dv/dt so printable nursing report sheet pdf differentiate the above equation lincoln university email address respect to t: a = dv/dt = 4. The acceleration is constant at feminist approach in research methodology 2 . If there is a relationship between energy (E) and time (t), research paper on economic growth in india the derivative of E with respect to t, dE/dtgives the power (P) at any time. Example teenager before and now essay A device uses up energy in a manner dependent on time: E = t 3where E printable nursing report sheet pdf energy in Joules and critical thinking brooke noel moore 12th edition is the time bikini climber reportedly freezes seconds. Find the power being used after 2s. The power, P, is given by dE/dt printable nursing report sheet pdf we differentiate the above equation with respect to t: P = dE/dt = 3t 2. The power after 2s is therefore 12W. Differentiation is thus one of the most powerful tools in mathematics and physics.