So, (1) Any line with positive slope through (2,1) meets the condition for example f (x) = 3(x − 2) 1 has f '(x) = 3 > 0 for all x ≠ 2 and also for x = 2 graph {y1=3 (x2) 1907, 919, 245, 31} Replace 3 with any positive number for another exampleF x = x ^ { 2 } 6 x 8 f x = x 2 − 6 x 8 The equation is in standard form The equation is in standard form xf=x^ {2}6x8 x f = x 2 − 6 x 8 Divide both sides by x Divide both sides by x \frac {xf} {x}=\frac {\left (x4\right)\left (x2\right)} {x} HSBC244 has shown a nice graph that has derivative #f'(3)=0# Here are couple of graphs of functions that satisfy the requirements, but are not differentiable at #3# #f(x) = abs(x3)5# is shown below graph{y = abs(x3)5 14, 25, 616, 1185} #f(x) = (x3)^(2/3) 5# is shown on the next graph
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What does f'(x) 0 mean- For x < 0 , f(x) = 1 – x We find the points to be plotted when x < 0 For x = 0 , f(x) = 1 Hence, point to be plotted is (0,1) For x > 0 , f(x) = x 1 We fi (टीचू) Maths† x = a is a minimum if f0(a) = 0 and f00(a) > 0;
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How to tell where f(x) greater than 0 or f(x) less than 0 How to tell where f(x) greater than 0 or f(x) less than 0(a)Find the derivative f0(x) lim h!0 f(x h) f(x) h = lim h!0 4 xh 4 x h = lim h!0 4x 4(xh) x(xh) h = lim h!0 4h hx(x h) = lim h!0 4 x(x h) = 4 x2 (b)Find and interpret f0(5) f0(5) = 4 52 = 4 25 The slope of f at x = 5 is 4 25 (c)When x = 5, is the graph of f(x) increasing, decreasing, or neither?F(x) = (c) Use calculus to find the exact maximum value of f(x) f(x) = (d) Use a graph of f '' to estimate the xcoordinates of the inflection points (Round your
0 (b) Explain the shape of the graph by computing the limit as x ?GRAPH OF f(x) = x^3 f(x) = x^3 graph how to draw graph of f(x)=x^3 y = x^3 graphNCERT CLASS 11 MATHS SOLUTIONS ex 23SUBSCRIBE TO MY CHANNEL TO GET MORE UPDACos x about x =0 Use this series and the series for sin ,()x2 found in part (a), to write the first four nonzero terms of the Taylor series for f about 0x = (c) Find the value of f ()6 ()0 (d) Let Px4( ) be the fourthdegree Taylor polynomial for f about 0x = Using information from the graph of yf x= ()5 shown above, show that ( ) ( ) 4 11
Steps for Solving Linear Equation f ( x ) = 1 \frac { 2 } { x 1 } , s f ( x) = 1 − x 1 2 , s Multiply both sides of the equation by x1 Multiply both sides of the equation by x 1 fx\left (x1\right)=x12 f x ( x 1) = x 1 − 2 Use the distributive property to multiply fx by x1 Transcript Ex 51, 8 Find all points of discontinuity of f, where f is defined by 𝑓(𝑥)={ (𝑥/𝑥, 𝑖𝑓 𝑥≠0@&0 , 𝑖𝑓 𝑥=0)┤ Since we need to find continuity at of the function We check continuity for different values of x When x = 0 When x > 0 When x < 0 Case 1 When x = 0 f(x) is continuous at 𝑥 =0 if LHL = RHL = 𝑓(0) Since there are two differentThe expression f(0) represents the yintercept on the graph of f(x) The yintercept of a graph is the point where the graph crosses the yaxis This
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Let us start with a function, in this case it is f(x) = x 2, but it could be anything f(x) = x 2 Here are some simple things we can do to move or scale it on the graph We can move it up or down by adding a constant to the yvalue g(x) = x 2 C Note to move the line down, we use a negative value for C C > 0 moves it up;F 0(x) = k f (x) Solution Multiply the equation above f 0(x) − kf (x) = 0 by e−kx, that is, f 0(x) e−kx − f (x) ke−kx = 0 The lefthand side is a total derivative, f (x) e−kx 0 = 0 The solution of the equation above is f (x)e−kx = c, with c ∈ R Therefore, f (x) = c ekx C7 Consider the following 1 x 1 lim x x 1 Use the graph to find the limit (if it exists)
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Δx 0 answer We've computed −that f (x) = 1 Is this correct?If f′(x) > 0 for all x ∈(a,b), then f is increasing on (a,b) If f′(x) < 0 for all x ∈(a,b), then f is decreasing on (a,b) First derivative test Suppose c is a critical number of a continuous function f, then Defn f is concave down if the graph of f lies below the tangent lines to f Defn f is concave up if the graph of fF0(x) = (2x if x>0;
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0 if x 0 For x>0, the derivative is f0(x) = 2xas above, and for xFor example, if f is a function that has the real numbers as domain and codomain, then a function mapping the value x to the value g(x) = 1 / f(x) is a function g from the reals to the reals, whose domain is the set of the reals x, such that f(x) ≠ 0 The range of a function is the set of the images of all elements in the domainHow might we check our x2 0 work?
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A function may be thought of as a rule which takes each member x of a set and assigns, or maps it to the same value y known at its image x → Function → y A letter such as f, g or h is often used to stand for a functionThe Function which squares a number and adds on a 3, can be written as f(x) = x 2 5The same notion may also be used to show how a function affects particular valuesAnswer to Graph f(x) = 3^x 1 By signing up, you'll get thousands of stepbystep solutions to your homework questions You can also ask yourThis is an absolute value function It forms a v shaped graph For positive xvalues, and zero, it is a line with a yintercept of zero and a slope of one y = x if x > 0 or x = 0 For negative xvalues it is a line with a yintercept of zero and a slope of negative one y = x if x < 0
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Let f be continuous on an open interval (a,b) that contains a critical xvalue 1) If f'(x) > 0 for all x on (a,c) and f'(x)0 for all x on (c,b), then f(c) is a local maximum value 3) If f'(x) has the same sign on both sides of c, then f(cSince we're looking for f(x)=0, we're looking for y=0 since y and f(x) can be interchanged In other words, we are looking for the xintercept, since y=0 for all xintercepts So we substitute 0 in for f(x) and we get Now we solve for x Add 12 to both sides Divide both sides by 3 This will isolate xConsider the function below f(x) = x1/x, x >
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C < 0 moves it downFor the first case, we see that f(x) = 0 will solve our original function, since R x 0 0dx = 0 for all x In the second case, f0(x) = 1 2, so f(x) = 1 2 x C To get the value of C, notice in the original equation that if x = 0, then Z 0 0 f(x)dx = (f(0))2 ⇒ f(0) = 0 Thus, C = 0 So, we have two possibilities f(x) = 0 or f(x) = 1 2 x 30 For f ( x) ∗ f ′ ( x) < 0 either f ( x) is positive and f ′ ( x) is negative or vice versa I can see if f ( x) > 0 by seeing if the graph at x is a above or below the xaxis I can see if f ′ ( x) > 0 by seeing if at x the graph is increasing or not So I look for the areas of the graph with it is either both above the xaxis and decreasing or below the xaxis and increasing
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Graph f (x)=0 f (x) = 0 f ( x) = 0 Rewrite the function as an equation y = 0 y = 0 Use the slopeintercept form to find the slope and yintercept Tap for more steps The slopeintercept form is y = m x b y = m x b, where m m is the slope and b b is the yintercept y = m x b y = m x b Find the values of m m and b b using theSolution set of 'f(x)=0' xintercept(s) of graph Solution set of 'f(x)>0' xvalues corresponding to part of graph ABOVE xaxis Free, unlimited, online practiceFirst of all, f (x 0) is negative — as is the slope of the tangent line on the graph of y = 1 Secondly, as x 0 →∞ (ie as x 0 grows larger and larger), the x tangent line is less and less steep So x 1 should get closer to 0 as x 0
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Go to your Math tools and select the Graph tool to graph this situation Set the window so that it shows yvalues from 0 to 100 and xvalues from 0 to 10 Then, paste a copy of the graph in the space provided4321 0 1 2 3 44 3 2 1 0 1 2 3 4 The graphs of f(x) = sinx, and y = x,y = x− x3 6, y = x− x3 6 x5 1,y = x− x3 6 x5 1 − x7 5040X is not equal to 0 or 1 Please explain how to get all values of x when f(x) > 0 and f(x) < 0 and explain how to graph
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Since the input \(x\) can be any real number the range of \(f\) is all the integers, \(\mathbb{Z}\) The function \(g(x)=xx\) which means it subtracts the whole number part, leaving only the fractional part of the input value \(x\) For integer values of \(x\), \(x=x\) which means that \(g(x)=0\) So the graph of the function looks like thisF(x) = abx where a and b are constants, b > 0 and b ≠ 1 The independent variable is in the exponent Ex f(x) = 2x is an exponential function, but f(x) =Sketch the graph of one function that satisfies ALL of the following conditions (Many correct answers are possible) f(0) = 0 f(4) = 5 So far Sec 26 Limits at Infinity We have seen that the limit of a function at x = a may be ∞ or ∞ But it is also possible to find a limit at infinity
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If you were to get the slope of f (x) at the far left it would be increasing 1, 2, 3 and peaking at 4 around x = 75 Then decreasing to 3, 2, 1, 0 Note that those decreasing values 3, 2, 1 are still positive Again its not easy to see just looking at f (x) but the graph of f ' (xTranscribed image text Find all values of x such that f(x) > 0 and all x such that f(x) < 0 and sketch the graph of f 1) F(x) = x^2(x2)(x1)^2(x2) Answer f(x) > 0 if x > 2 f(x) < 0 if x < 2 ;Xaxes y = 0 The range of a function f is the set of values which the function f takes The function f(x,y) = 1 x2 for example takes all values ≥ 1 The graph of f(x,y) is the set {(x,y,f(x,y)) (x,y) ∈ D } The graph of f(x,y) = p x 2y2 on the domain x y2 < 1 is a half sphere Here are more examples
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From the graph of f(x), draw a graph of f ' (x) We can see that f starts out with a positive slope (derivative), then has a slope (derivative) of zero, then has a negative slope (derivative) This means the derivative will start out positive, approach 0, and then become negative Be Careful Label your graphs f or f ' appropriately When we're graphing both functions and their derivativesA logarithmic function f(x) = log a (x) , a > 0, a 1, x > 0 (logarithm to the base a of x) is the inverse of the exponential function y = ax Therefore, we have the following properties for this function (as the inverse function) (I) y = log a (x) if and only if a y = x This relationship gives the definition of log a (x f(0) = 1 ≥ 1 x2 1 = f(x) for all real numbers x, we say f has an absolute maximum over ( − ∞, ∞) at x = 0 The absolute maximum is f(0) = 1 It occurs at x = 0, as shown in Figure 412 (b) A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither
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A function f has an absolute maximum at c if f(c) ≥ f(x) for all x in the domain of f The function value f(c) is the maximum value A function f has an absolute minimum at c if f(c) ≤ f(x) for all x in the domain of f The function value f(c) is the minimum value The absolute maximum and minimum values are called the extreme values of fExplain why Since 4 25 is positiveA) If f'(x) >0 on an interval, then f is increasing on that interval b) If f'(x) 0 on an interval, then f is concave upward on that interval d) If f''(x)
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0 and as x ?2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all x For this equation, The parabola opens upward because a > 0, resulting in a vertex that is a minimum The yintercept of the quadratic function f (x) = x² 3x 4 is (0, c), ie, the point From we get the xintercepts at The axis of symmetry is , ie, The minimum value is The vertex is where or
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01 Reminder For a function of one variable, f(x), we flnd the local maxima/minima by difierenti ation Maxima/minima occur when f0(x) = 0 † x = a is a maximum if f0(a) = 0 and f00(a) < 0;A point where f00(a) = 0 and f000(a) 6= 0 is called a point of in°ection Geometrically, the equation y = f(x) represents a curve in the twoAlgebra Graph f (x)=x f (x) = x f ( x) = x Find the absolute value vertex In this case, the vertex for y = x y = x is (0,0) ( 0, 0) Tap for more steps To find the x x coordinate of the vertex, set the inside of the absolute value x x equal to 0 0 In this case, x = 0 x = 0 x = 0 x = 0
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