That is, the rule for this transformation is –f (x) To see how this works, take a look at the graph of h(x) = x 2 2x – 3F ix i 2 Evaluate X4 k=1 k2 2 Some examples Example Evaluate X4 r=1 r3 Solution This is the sum of all the r3 terms from r = 1 to r = 4 So we take each value of r, work out r3 in each case, and add the results Therefore X4 r=1 r3 = 13 23 33 43 = 1764 = 100 Example Evaluate X5 n=2 n2 Solution In this example we have used the letter n to represent the variable in the sum, rather0Given that x >
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51 Maxima and Minima A local maximum point on a function is a point (x, y) on the graph of the function whose y coordinate is larger than all other y coordinates on the graph at points close to'' (x, y) More precisely, (x, f(x)) is a local maximum if there is an interval (a, b) with a <đạo hàm trên \(\mathbb{R}\) làOr e x can be defined as f x (1), where f x R → B is the solution to the differential equation df x / dt (t) = x f x (t), with initial condition f x (0) = 1;
Solutions to Graphing Using the First and Second Derivatives SOLUTION 1 The domain of f is all x values Now determine a sign chart for the first derivative, f ' f ' ( x) = 3 x2 6 x = 3 x ( x 2) = 0 for x =0 and x =2 See the adjoining sign chart for the first derivative, f ' Now determine a sign chart for the second derivativeSolution Steps f ( x ) = x ^ { 4 } ( x 1 ) ^ { 3 } f ( x) = x 4 ( x − 1) 3 Use binomial theorem \left (ab\right)^ {3}=a^ {3}3a^ {2}b3ab^ {2}b^ {3} to expand \left (x1\right)^ {3} Use binomial theorem ( a − b) 3 = a 3 − 3 a 2 b 3 a b 2 − b 3 to expand ( x − 1) 3And, as in the first example, =!
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F(x) = ˆ cx4 0 <x<1 0 otherwise Find a E(X) b Var(X) Example 8 To be a winner in the following game, you must be succesful in three succesive rounds The game depends on the value of X, a uniform random variable on (0,1) If X>01, then you are succesful in round 1;Using our notation above, f(x) = x 2 f(1) = 1, f(2) = 4, f(3) = 9, f(4) = 16, f(5) = 25, f(6) = 36 P(X = 1) = 1/6, P(X = 2) = 1/6, etc So E(X 2) = 1/6 4/6 9/6 16/6 25/6 36/6 = 91/6 = The expected value of a constant is just the constant, so for example E(1) = 1 Multiplying a random variable by a constant multiplies the expected value by that constant, so E2X = 2EX ACounselling Sessions to those students who faced difficulties in Maths
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Explanation Let f (x) = 12xx^34x^5 and note that for every x, x is a root of the equation if and only if x is a zero of f f has at least one real zero (and the equation has at least one real root) f is a polynomial function, so it is continuous at every real number In particular, f is continuous on the closed interval 1,0Professionals For math, science, nutrition, historySafety How works Test new features Press Copyright Contact us Creators
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The cumulative distribution function (CDF) of random variable X is defined as FX(x) = P(X ≤ x), for all x ∈ R Note that the subscript X indicates that this is the CDF of the random variable X Also, note that the CDF is defined for all x ∈ R Let us look at an example Example I toss a coin twice Let X be the number of observed headsPurplemath The last two easy transformations involve flipping functions upside down (flipping them around the xaxis), and mirroring them in the yaxis The first, flipping upside down, is found by taking the negative of the original function;Professionals For math, science, nutrition, history
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0, Multiplying 3 both sides 3x >The function f is defined by f (x) = x4 −4x2 x 1 for −5 ≤ x ≤ 5 What is the interval in which the minimum of value of f Purely a graphical approximation;F(4) = p(0) p(1) p(2) p(3) p(4) = 1 16 4 16 6 16 4 6 1 16 = 16 16 We can write this in an alternative fashion as 4 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS FX(x)= 0 forx <0 1 16 for0 ≤ x<1 5 16 for1 ≤ x<2 11 16 for2 ≤ x<3 15 16 for3 ≤ x<4 1 forx≥ 4 164 Second example of a cumulative distribution function Consider a group of N individuals, M of whom are
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For the other equation, y = (1/4)x 2, going from the point (3, 9/4) to (4, 4), we would rise only 1 &Knowledgebase, relied on by millions of students &About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy &
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Solution For For x\epsilon R \left \{0, 1\right \}, let f_{1}(x) = \dfrac {1}{x}, f_{2}(x) = 1 x and f_{3}(x) = \dfrac {1}{1 x} be three given functions If aF(x) = cos( 0 ) i sin( 0 ) = 1 g(x) = C 3 e i 0 = C 3 These functions are equal when C 3 = 1 Therefore, cos( x ) i sin( x ) = e i x Justification #2 the series method (This is the usual justification given in textbooks) By use of Taylors Theorem, we can show the following to be true for all real numbers sin x = x x 3 /3!Knowledgebase, relied on by millions of students &
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Extended Keyboard Examples Upload Random Compute answers using Wolfram's breakthrough technology &0Multiplying 1 both sides – 1 ×The function PX(xk) = P(X = xk), for k = 1, 2, 3,, is called the probability mass function (PMF) of X Thus, the PMF is a probability measure that gives us probabilities of the possible values for a random variable While the above notation is the standard notation for the PMF of X, it might look confusing at first
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2 0 (We need to make it in formFirst, go to the point P (x3, f(x3)) P ( x 3, f ( x 3)) on the graph of y = f(x) y = f ( x) This point has the y y value that we want, but it has the wrong x x value Move this point 3 3 units to the left Thus, the y y value stays the same, but the x x value is decreased by 3 3Ex 23, 5 Find the range of each of the following functionsf(x) = 2 – 3x, x ∈ R, x >
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F (g (2)), g (x)=2x1, f (x)=x^2 \square!Compute answers using Wolfram's breakthrough technology &3/4 or 7/4, which is less than 7 So, the second parabola is broader than the first parabola as illustrated in the graph below Graph of the parabolas, y = x 2 (blue) and y = (1/4)x 2 (red) The general charactersitics of the value a, the coefficient
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About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy &Professionals For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music WolframAlpha brings expertlevelF'(x) = (2x3)^3(x^2 x 1)^4(28x^212x7) Looking at the equation, f(x) = (2x3)^4(x^2x1)^5 we first notice a couple patterns 1 The function is a product of two terms 2 Each of the terms is a term with an exponent Since the function is a product of two terms, we know that we have to use the Product Rule to find the first derivative
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If X>02, then you are succesful in round 2;For example, f (x) = e −1/x 2 can be written as a Laurent series Generalization Since the cosine is an even function, the coefficients for all the odd powers x, x 3, x 5, x 7, have to be zero Second example Suppose we want the Taylor series at 0 of the function = We have for the exponential function = =!Since (x1)(x4) = 0 for x=1 and x=4 , function f is continuous for all values of x EXCEPT x=1 and x=4 Click HERE to return to the list of problems SOLUTION 7 First describe function g using functional composition Let f(x) = x 1/3, , and k(x) = x 5 Function k is continuous for all values of x since it is a polynomial, and functions f and h are wellknown to be continuous for all
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Knowledgebase, relied on by millions of students &For example, let f(x) = 6x 4 − 2x 3 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity This function is the sum of three terms 6x 4, −2x 3, and 5 Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x 4 Now one may apply the second rule 6x 4 isUse the distributive property to multiply 3x^{4}18x^{3}24x^{2} by x1 and combine like terms Use the distributive property to multiply 3 x 4 − 1 8 x 3 2 4 x 2 by x 1 and combine like terms 3x^{5}15x^{4}6x^{3}24x^{2} 3 x 5 − 1 5 x 4 6 x 3 2 4 x 2 Expand 3x^{5}15x^{4}6x^{3}24x^{2} 3 x 5 − 1 5 x 4 6 x 3 2 4 x 2 View solution steps Solution Steps f ( x ) = 3 x ^ { 2
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0Adding 2 both sides 2 – 3x <So, the inverse of f(x) = 2x3 is written f1 (y) = (y3)/2 (I also used y instead of x to show that we are using a different value) Back to Where We Started The cool thing about the inverse is that it should give us back the original value When the function f turns the apple into a banana, Then the inverse function f1 turns the banana back to the apple Example Using the formulas fromX^2(y(x^2)^(1/3))^2 = 1 Natural Language;
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F(x 1) = 3(x 1) 4 = 3x 7 Domain and Range The domain of a function is the set of values which you are allowed to put into the function (so all of the values that x can take) The range of the function is the set of all values that the function can take, in other words all of the possible values of y when y = f(x) So if y = x 2, we can choose the domain to be all of the real numbersMinimum f = 463, nearly This is improved to 8sd, \displaystyle {} , using an iterative numerical methodB and f(x) ≥ f(z) for every z in both (a, b) and
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Consider x^ {2}4x3 Factor the expression by grouping First, the expression needs to be rewritten as x^ {2}axbx3 To find a and b, set up a system to be solved a=3 b=1 Since ab is positive, a and b have the same sign Since ab is negative, a and b are both negative The only such pair is the system solutionKnowledgebase, relied on by millions of students &Beyond simple math and grouping (like (x2)(x4)), there are some functions you can use as well Look below to see them all They are mostly standard functions written as you might expect You can also use pi and e as their respective constants Please note You should not use fractional exponents For example, don't type x^(1/3) to compute the cube root of x Instead, use root(x,3
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If x=7 4√3 and xy=1 then 1/x^2 1/y^2=I'm providing Free Online Live Maths Classes &Get stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!The output f (x) is sometimes given an additional name y by y = f (x) The example that comes to mind is the square root function on your calculator The name of the function is \sqrt {\;\;} and we usually write the function as f (x) = \sqrt {x} On my calculator I input x for example by pressing 2 then 5 Then I invoke the function by pressing
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Integration by substituting u = axb 2 3 Finding Z f(g(x))g So, substituting u for 3x4, and with dx = 1 3 du in Equation (2) we have Z cos(3x4)dx = Z 1 3 cosudu = 1 3 sinuc We can revert to an expression involving the original variable x by recalling that u = 3x 4, giving Z cos(3x4)dx = 1 3 sin(3x4)c We have completed the integration by substitution It is very easy to generaliseF (x) is decreasing when x = −1 Explanation To determine whether f (x) is increasing or decreasing at x = −1 More Items x^ {2}3x2x64 Apply the distributive property by multiplying each term of x2 by each term of x3 x^ {2}x64 Combine 3x and 2x to get x x^ {2}x2 Add 6 and 4 to get 2
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