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AB/BC Summer Calculus Assignment Answers ExplainedIris GalfasIf there are any errors with these problems please notify me so I can fix them.2x+3y=8 and x+2y=5x+2y=5x=5-2y2x+3y=825-2y+3y=810-4y+3y=8y=2x+22=5x=1y=x2+2x+9 and 7x+y=197x+y=19y=19-7xy=x2+2x+919-7x=x2+2x+9x2+9x-10=0x+10x-1=0x=-10 and x=17x+y=197-10+y=19 and 7(1)+y=19y=89 and y=12The length, l, of a certain rectangle is twice the width, w. Write an equation for the perimeter the rectangle as a function of the width, w. l=2wP=2l+2wP=22w+2wP=6wIf the area of the rectangle described above is 50 square feet, find the length and the width of the rectangle.A=lw50=lwl=2w50=2ww50=2w225=w2w=±5Only positive values make sense for a lengthw=5l=2wl=25l=10Find the point of intersection between the lines y=x+1 and 3y-x=53y-x=53x+1-x=53x+3-x=52x=2x=1y=x+1y=1+1y=2Find the point of intersection between y=x+7 and y=x2+2x+5x+7=x2+2x+5x2+x-2=0x+2x-1=0x=-2 and x=1y=-2+7 and y=1+7y=5 and y=8a point is in the first quadrant when both the x and y values of the point are positivef2=5 means that if 2 is the value put into the function f(x) then 5 will be the value returnedan expression is a function if and only if for every x value there is only one y value (a function is one-to-one if for every y value there is only one x value) a zero of a function is 4 if the function crosses the x axis at the value of four (the zero of a function is also called the x-intercept and the root of the function and finding the solutions to a function is the same as finding the zeros)y is directly proportional to x if as x increases y also increases, example: y=2xthe coefficient of the third term is 5 when the number before the variable in the third term of an expression is 5, example: 2x3+7x2+5x+4a function has only one root if it crosses the x-axis only once“a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents” (definition from Wikipedia, I couldn’t find a way to define it that made sense)two triangles are similar if all corresponding angles have the same measure and the sides of one triangle can be found by multiplying the sides of the other triangle by the same numbera function is even if it has symmetry across the y-axisa function is odd if it has symmetry around the originthe quadratic formula is x=-b±b2-4ac2athe Pythagorean Theorem is a2+b2=c2the hypotenuse of a 45-45-90 isosceles right triangle with a leg of length x is x2the legs of a 45-45-90 triangle are equala2+b2=c2x2+x2=c22x2=c2x2=cthe hypotenuse of a 30-60-90 right triangle with shortest leg having a length of x is 2xthe short leg of a 30-60-90 triangle is always half the length of the hypotenusethe volume of a sphere is 43πr3the volume of a cylinder is πr2h the volume is the area of a circle (the base of the cylinder) πr2 times the height h of the cylinderthe volume of a cone is 1/3 the volume of a cylinder so 13πr2h is the volume of a conethe volume of a box with a square base is the area of the square that is the base times the height of the box so the volume is x2hthe surface area of a sphere is 4πr2 this takes integrals to prove but the basic idea is that that the SA of a sphere is the SA of a cylinder with open top and bottom that fits around the sphere perfectly, so the cylinder would have a height of 2r and a radius of rthe surface area of a cylinder with no top is πr2+2πrh the SA of the circle that is the base times the SA of the side of the cylinderthe area of a triangle is 12bh if you were to create a rectangle around the triangle with a base and height the same as the triangle’s the area of the rectangle would be bh and twice the triangle’sthe area of a trapezoid is b1b22h half of the triangles on the ends of the trapezoid can be cut off and rotated 180° to fit with the half left and create a rectangle with a base of the average of the bases of the trapezoid and the same height. This rectangle has the same area as the trapezoidthe cross section through the center of a sphere is πr2. if a sphere has a radius of r then the largest circle that can be created with a cross section of it will have an area of πr2 and will be cut through the center of the spherethe area of an equilateral triangle in terms of the length of a side s is s234. The base of the triangle will be s and the height can be found using the Pythagorean theorem by dividing the triangle into two right triangles by drawing an altitude aa2+(s2)2=s2a2=s2-s22a2=4s24-s24a2=3s24a=±s32A negative length has no meaning in this problem so we throw out the negative solution. The area can now be found by substituting the side lengths into the equation for the area of a triangle 12bhA=12ss32A=s234A six foot man is standing 10 feet away from a 20 foot lamppost. What is the length of his shadow? This problem can be solved using similar triangles. A diagram is helpful (not drawn to scale)x+1020=x66x+10=20x6x+60=20x60=14x307=xWater is dripping out of a conical figure that has a diameter of 8 inches and a height of 12 inches. When the depth of the water is 8 inches, what is the radius of the water? This is another similar triangles problem. A diagram is helpful (not drawn to scale)x8=412x=83Find the equations of the horizontal and vertical asymptotes y=1x-1 V.A.:x=1 H.A.:y=0The vertical asymptote is at x=1 because that x value will cause the denominator of the function to equal zero. Since dividing by 0 is not allowed the function doesn’t exist at x=1 and there is an asymptote. The horizontal asymptote is at y=0 because the value of the function gets very close to 0 as x gets very small or large, but can never reach it.Exponent Rules: Which of the following are true?x0=1 True. Exponents say multiply the base, x, by itself the number of times as the exponent. For example x2=x*x andx3=x2*x=x*x*x. Each time the exponent increases by 1 the number of times x is multiplied by itself also increases by 1. If the exponent is decreased by 1 then the result would be multiplied 1 less time by x or divided by x. For example x2=x3x andx=x2x. If this is extended to x0 then we find thatx0=x1x. x1=x and dividing something by itself results in 1 so x0=1x-2=1x2 True. When expression with the same base and exponents are divided the exponents are subtracted. The equation x-2=x0x2 is the same as the first equation but it shows how the exponents are subtracted better. 0-2=2x+y=x+y FALSE. Try this with numbers and you will see that it is not true. 16+9≠16+925≠4+35≠7x5*x3=x15 FALSE. Add exponents when multiplying, do not multiply them x5*x3 should equal x158x5*y5=xy5 True. The exponent says multiply everything it acts on by itself the number of times as the exponent. Because the exponent is distributed to all the things in the parenthesis the expression is true. xy5=x*y5 is false. x35=x8 FALSE. When raising an exponent to an exponent multiply the exponents, do not add them. x35should equal x15x5-w=x5xw True. Subtracting numbers in exponents equals dividing the base raised to the first number by the base raised to the second number.xt+5=xt5 FALSE. Adding exponents equals multiplying the base raised to the first number by the base raised to the second number. xt+5should equal xt*x594=32 True. Make sure to square root all the things inside the radical4x12=2x FALSE. Make sure to raise all the things in the parenthesis to the power of the exponent. 4x12 should equal 2x 1x=x-12 True. The square root of x is x12 and 1x=x-1. Combining we get x-12x2=x FALSE. The square root of something always has both positive and negative parts. x2 equals the absolute value of x or ±xx2-25=x-5 FALSE. Try this with numbers and you will see that it is not true.62-25≠6-511≠1x43=4x3 FALSE. The numerator of an exponent is the power and the denominator is the root. x43 should equal 3x4 x12+y122=x+y FALSE. Try this with numbers and you will see that it is not true.1612+9122≠16+94+32≠2549≠25x-23=13x2 True. When an exponent is negative it means to take the inverse of the base or 1 divided by the base. The numerator of the exponent it the power the base is raised to and the denominator is the root of the base.elnx2=x2 True. Taking logx of something and raising x to the power of that something are inverse operations which undo each other.e2ln2-ln5=45 True. Subtraction in an exponent can be represented with division e2ln2-ln5=e2ln2eln5A number multiplied by log x equals log (x to the power of the number)e2ln2eln5=eln22eln5Taking logx of something and raising x to the power of that undo each other. eln22eln5=45lnx2=lnx2 FALSE. When you have the log of a number to a power the power only acts on the number, not the whole function. lnx2 should equal 2lnxExpand using the properties of logarithms: ln33x+74x+1035x-8213ln3x+74x+1035x-82 The power of what you are taking the log of can be rewritten as multiplied by the log13(ln3x+74+lnx+103-ln5x-82) Multiplication inside the log can be rewritten as addition and division as subtraction13(ln4(3x+7)+ln3(x+10)-ln2(5x-8) Same as the first stepCondense into a single logarithmic expression using the properties of logarithms: 17lnx-23ln(x5+5)lnx17-ln3x5+52lnx173x5+52Using a Graphing Calculator:Graph 0.1x3+2x2-x-3 Graph from Desmos (if you haven’t seen it Desmos is a great online graphing calculator and you should check it out)Find the roots of the equation. -20.418, -1.021 and 1.439. Roots are the same as solutions and zeros. You will have to read your calculator’s manual to find out how to do this. One tip to make sure not to leave out ant roots when searching for them is to have an idea of how many there are. The maximum number of root an equation can have is equal to the highest power of x, for example if the equation is a quadratic, so x2 is the highest power of x, then the maximum number of roots to the equations is 2. If the function only touches the x-axis but does not cross it then this counts as 2 roots.Find the point of intersection for the graphs y=x3+x-3 and y=2x+4 (2.087, 8.173) You will need to read your calculator’s manualFind the maximum value for the graph fx=-x4+x-4 -3.528 You will need to read your calculator’s manualFor the function in #151, find the intervals on which fx is increasing.(-∞, 0.630) You will need to read your calculator’s manual. First find all the relative maximums and minimums of the graph (thins graph only has 1 maximum point) then determine when the graph is increasing within the intervals created in between the maximums and minimumsWhat are the following trigonometric identities?secx=1/cosx This is the definition of secxcscx=1/sinx This is the definition of cscx opposite adjacent hypotenusetanx=sinx/cosxsinx=opposite hypotenusecosx=adjacenthypotenusetanx=oppositeadjacenttanx=sinxcosx=opposite hypotenuseadjacenthypotenusetanx=opposite hypotenuse*hypotenuseadjacent=oppositeadjacentcotx=cosx/sinx cotx is the inverse of tanxcos2x-1=-sin2xx2+y2=r2 The Pythagorean Theorem applied to the unit circle where x is the length of the adjacent side of the triangle, y in the length of the opposite side and r is the length of the hypotenusex2r2+y2r2=r2r2sin2x+cos2x=1cos2x-1=-sin2xsec2x-1=tan2xThis can be shown using the same setup as the last problemx2+y2=r2x2x2+y2x2=r2x21+tan2x=sec2xsec2x-1=tan2xcot2x+1=csc2xThis can be shown using the same setup as the last problemx2+y2=r2x2y2+y2y2=r2y2 cot2x+1=csc2xEvaluate the following expressions:The values of a trigonometric expression acting on an angle is the ratio of the sides of a right triangle with angle that the trigonometric expression is acting on. What sides are used in the ratio depend on which trigonometric function is being evaluated. sin is opposite/hypotenuse, cos is adjacent/hypotenuse, tan is opposite/adjacent, csc is 1/sin or hypotenuse/opposite, sec is 1/cos or hypotenuse/adjacent and cot is 1/tan or adjacent/opposite. The inverse trigonometric functions take the value of the relationship between the sides of a right triangle with some angle and ask you to find that angle. You should be able to find the trigonometric values for all the special angles without a calculator.sinπ6=12 cos-132=π6tan7π6=33cos0=1cosπ4=22csc-5π6=-2secπ=-1cot-π2=0sin-112=π6tanπ2=undefinedsin25π6=14cot2π3=-33sinπ2=1cot-1-1=3π4sec3π4=-2tan-1-1=-π4cscπ=undefinedsec2π4=2Sketch one period of the following trigonometric graphs (all graphs from Desmos)sinxcosxtanxsecxcscxcotxSolve the following trigonometric equation for the given domain: sinx=cosx on 0,2πThe only time sinx=cosx is when they both equal 22 or -22. This happens at x=π4 where both equal 22 and at x=5π4 where both equal -22. ................
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