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MathTo graph f(x) = 3sinx and g(x) = sin3x Take the pair of suitable values for f(x) and x and plot the pairs (x,3sinx). The pair of certain values are given below fro x=0 to 180. x ....= 00,...

MathWe'll change the base of the logarithm, so that: log16(24)=log3(24)/log3(16)=log3(2^3*3)/log3(2^4) We'll use the product property of the logarithms and also the power property:...

Mathf(x) = [(e^x)  (e^x)]/ (e^x)+(2^x)] = [(e^x)(1/e^x)]/[(e^x)+(1/e^x)] = (e^2x 1)/ (e^2x) +1) = (e^2x +1 2)/ (e^2x)+1) = 1 [2/(e^2x)+1] y= 1 [2/(e^2x)+1)] y1=...

MathTo prove this inequality, we'll use Lagrange's theoreme, for the function f(x)=sqrt x, where x belongs to the interval [64,68]. Let's recall Lagrange's theoreme, for a function f(x), on the...

Mathf(x) = 12x To find f(f(x)) we will substitute with x=f(x) ==> f(f(x))= 12(f(x) = 12(12x) = 12+4x = 1+4x The sum:\ S= 1+4(1) + 1 + 4(2) +...

MathA root has a multiplicity order only if it's a root for the function and for it's first derivative and for the second derivative and so on, considering the multiplicity order....

MathFor verifying if a function is bijective, we can calculate the first derivative of the function, to verify if the function is strictly increasing. If the function is strictly increasing, then the...

MathThis is clearly an arithmetic progression. There is the same interval between each of the times that the buses leave the station. Therefore, all you have to do is keep adding that interval to the...

MathFirst of all, we'll rearrange the expression given: a lna +b lnb<(a+b)*ln(a+b) We'll divide the inequality above, with the quantity (a+b) and the inequality will become: [a/(a+b)]*lna +...

MathIn order to prove that f(x) is concave, we have to demonstrate that f"(x) < 0; but for f"(x) calculus, we must calculate firs derivative, first. f'(x) = 1/(1 + x^2), f"(x) = [1'*(1 + x^2) ...

MathFor x =  1 to be a multiple root of order 3, it has to cancel the function, the first derivative, the second derivative,but the third one, does not. f(1) = 0, f'(1) = 0, f"(1) = 0 f(1) =...

Math4^x 2^x =56 4^x 2^x 56= 0 Now let us rewrite 4^x as (2^x)^2 Now assume that 2^x = y ==> y^2 y 56 =0 ==> (y8)(y+7) ==> y1= 8 ==> y1= 2^x1=8 ==> X1= 3 ==> Y2= 7 ==>...

Matha3 + a4 =8 r=2 We need to calculate a1 Since a1, a2, a2, a4 are terms of arithmetical progression, then: a2= a1+2....(1) a3= a1+ 4....(2) a4= a1+ 6.....(3) Now add (2) and (3): a3+ a4 = 2a1 + 10...

Mathx^2 3x+1 =0 ==> x^2 = 3x1 x1, and x2 are solutions, then f(x1)=0 and f(x2)=0 x1^2 =3x1 1 =0.....(1) x2^2 =3x2 1 =0 .....(2) Now add (1) and (2): x1^2 + x2^2 = 3(x1+x2) 2 But from Viete rule...

Mathsin a= 3/5 We know that sina = opposite/hypotenuse = 3/5 Then, the adjacent^2= 5^2 3^2 = 259=16 Then the adjacent= 4 Then cos a= adj./hypotenuse= 4/5. Now, we know that: tg(a/2)= sina / 1cosa...

MathIf the point A is in the second quadrant, that means that it's coordinates are: xA<0 yA>0 xA = t1 t1<0 We'll add 1 both sides: t<1 yA = t^2  3t t^2  3t>0 We'll calculate the...

MathThe formula for the circle is: x^2 + y^2 + ax + by +c =0 We have the triangle AOB , then A ,O and B are on the circle, Then A(4,2) , O(0,0) and B(2,4) should verify the equation: First we will...

Mathcos2a= 1/2 But we know that cos(2a) = cos^ a sin^2 a= 1/2 ==> sin^2 a= cos^2 a + 1/2.....(1) But sin^2 a + cos^2 a= 1 ==> sin^2 a= 1cos^2 a......(2) Now add (1) and (2): 2sin^2 a= 3/2...

MathIf ABCD is a parallelogram, then the diagonals should intersect in the midpoints. That means: midpoint of AC is the same midpoint of BD: The midpoint formula is ((x1+x2)/2 , (y1+y2)/2) midpoint for...

Mathf(x) = (m^2 2)x 3 If the function is decreasing, then the first derivative is negative, or f'(x)<0 Let us determine the derivative: f'(x) = m^2 2 m^22 <0 Add 2 ==> m^2 < 2 ==> m...

MathZ= 25/(4+3i) + 25/ (43i) First let us determine the common denominator: (43i)(4+3i)= 16 +12i12i 9(1)= 16 +9 = 25 Then : z= [25(43i) + 25(4+3i)]/25 Open brackets: z= (100  75i + 100...

MathFirst, let's substitute arctg (1/sqrt3) by the corresponding angle, namely pi/6. The equation will become: arctg(x/3) + pi/6 = pi/3 We'll subtract pi/6 both sides: arctg(x/3) = pi/3  pi/6 For...

Mathb1, 6, b3, 24 Since it is a geametric progression, then: 6= b1*r ==> b1= 6/r b3= 6r 24= 6*r^2 ==> r^2 = 24/6= 4 ==> r= 2 ==> b1= 6/r= 6/2=3 Let us check: b1, 6, b3, 24 3, 6, 12, 24...

MathWe'll open the first bracket: (2+i)(32i) = 6  4i + 3i  2i^2, where i^2=1 (2+i)(32i) = 6+2  i (2+i)(32i) = 8i We'll calculate the second bracket: (12i)(2i) = 2  i  4i + 2i^2 (12i)(2i)...

Mathmx +3y +2 =0 2x +ny 8 =0 If the lines are coincidental, then the ratios of the coefficients are equals. ==> m/2 = 3/n = 2/8= 1/4 ==> m/2 = 1/4 ==> m=1/2 ==> 3/n = 1/4 ==> n=...

MathWe'll solve the equation by applying the inverse function arcsine, to the right side of the equation: x  pi/4 = (1)^k*arcsin[sin(3x + pi/4)] + k*pi, where k is an integer number. But arcsin(sin...

Mathy= x^2 + 5x 6 We need to find the xintercept (P1) , and the yintercept (P2) The yintercept (P1) is (0,y) To find y, let us substitute with x=0: y= 0+06 =6 Then (yintercept) P1=(0,6) Now...

MathFirst, we'll use the property: log x (5) = 1/log 5 (x) Now, we'll rewrite the equation: log 5 (x) + 1/log 5 (x) = 5/2 We'll move all terms to one side: log 5 (x) + 1/log 5 (x)  5/2 = 0 We'll...

MathWe know that the hypotenuse of the right angle triangle, inscribed in a circle, is the diameter of the circle. So, all we have to do is to determine the hypotenuse of the right angle triangle,...

MathThe Greek letter pi is used to represent a universal constant that we come across many mathematical and scientific calculations. It is defined as the ration of the circumference of a circle to its...

MathWe'll consider z = a+b*i and it's conjugate z' = ab*i and we'll substitute them into the given relation. 2z'+z = 2(ab*i) + a+b*i 2z'+z = 2a  2bi + a + bi 2z'+z = 3a  bi But, from enunciation,...

MathFirst, we'll notice that we have to calculate a sum of 2 cubes, so we'll apply the formula: x1^3+x2^3 = (x1+x2)(x1^2  x1*x2 + x2^2) From Viete's relations, we'll ahve: x1 + x2 = 3 x1*x2 = 1 x1^2...

MathTo calculate the radius of a circle that is circumscribed to a triangle, we'll apply the formula: R = a*b*c/4*S, where: R  radius of the circle; a,b,c  sides of the triangle S  area of the...

MathTo calculate the coordinates of the common point, we have to solve the system formed by the equations of the line and parable. If the system does have real solutions, the values for x and y are the...

MathWe'll apply the formula: tg(a+b) = (tga + tgb)/(1tga*tgb) We also know that ctg x = 1/tg x. ctg a = 2, so tg a = 1/2 ctg b = 5, so tg b = 1/5 We'll substitute tg a and tg b by their values:...

MathTo prove that the line doesn't intercept the parable, that means that they don't have any common point, we have to solve the system formed by the equations of the line and parable. If the system...

MathFirst, we'll factorize: 2^x(1+2+1/2) = 56 We'll calculate the sum from the bracket: 1+2+1/2 = 3 + 1/2 = 7/2 The equation will become: 2^x * (7/2) = 56 2^(x1) * 7 = 7*8 We'll divide by 7: 2^(x1) =...

Math(5/(x9) + (2/(x+9) First we will determine the common denominator which is (x9)(x+9) = x^2 81 ==> [5(x+9) + 2(x9)]/(x^281) Now open brackets: ==> (5x+45 +2x 18)/(x^281) ==> (7x...

MathI will give you examples on how to right fractions using negative exponents" For example we have the fraction: 16/9 This could be written as: 4*4 / 3*3 = 4^2/3^2= (4/3)^2 If you flip the function ,...

MathHope you are writing 7 1/2 feet for 7 and 1/2 feet., and not 71/2 = 37.5feet. 7 and 1/2 feet = 7.5 feet = 12*7.5 = 90 inch. 9 inc = 0.75 feet. Therefore 90 inch wood you can make 9'' block of...

MathThe distance (d)= 1 1/3 = 4/3 mile Aaron runs 2/5 of the distance (d) ==> The distance he run = (2/5)*(4/3) = 8/15 mile The distance remains to Curtis house =...

MathLet the number be x: Then the quotient of the number and nine is x/9 Then the expression would be : y= (x/9) 3

MathUse a fifty cent piece, one quarter, three nickels, and five pennies. That gives you 10 coins, and a total of ninety five cents. You also cannot make change for a quarter, because the smaller coins...

Math(1+i)^20 = (1+i)^2*10 = [(1+i)^2 ]^10 = (2i)10 = (2^10)(i)^10 = (2^10)*(1) = (2^10)= 1024

MathS= sin(pi/3) + sin(2pi/3) + sin(3pi/3) + sin(4pi/3) Let us calculate each term: sin(pi/3)= sin (180/3)= sin(60)= sqrt(3)/2 sin (2pi/3)= sin(2(60)= sin(120)= sqrt(3)/2 sin(3pi/3)= sin(3(60)=...

Mathwe have : (z'+7i)/z=6 Let z= x+yi ==> z'= xyi Now substitute: (z'+7i)/z= 6 (xyi +7i)/x+yi =6 [xyi +7i]/(x+yi)=6 Multiply by z= (x+yi) ==> (xyi +7i]= 6(x+yi) ==> xyi +7i= 6x +6yi...

Mathsin(11pi/12) Let us rewrite 11pi/12 = 12pi/12 pi/12= pipi/12 ==> SIN(11PI/12)=sin (pipi/12) But, sin pix= sinx ==> sin(pipi/12)= sin(pi/12)= ==> sin(pi/12)= sin(1/2)*(pi/6) But we...

Math1) 10404 First we notice that the endign number (4) is even, then the number is divisible by 2: 10404= 2* 5202 Now also, 5202 is ending with an even number (2), then it is diisible by 2: 10404=...

MathA correlation describes how 2 VARIABLES are related and its strength. There are two types of variables, independent and dependent. So, from here you can eliminate your answer to either A or C. In...

MathA term in an agebraic expression could be a number , variable or both with any mathematical operation except + or minus. An algebraic expression wiith two or more terms are separated by + or...