MATH 41 | ULTIMATE REVIEW (Ch. 1-3, 10.2, 10.3, 10.4)
If f(x)=x²-4 and g(x)=√3-x, find (f/g)(-6)
(*Ans*) 32/3
Write the logarithmic function log₇343=3 in exponential form.
(*Ans*) 343=7³
Find the average rate of change of function given by g(x)= x²-6x+5, from x₁=-1 to x₂=3
(*Ans*) 4
Solve the inequality. (x+2)/(x-4)≥3
(*Ans*) 4<x<7
If f(x)=x²-4 and g(x)=√3-x, find (f o g)(-6)
(*Ans*) 5
Use properties of logarithms to simplify log₁₃13⁸.
(*Ans*) 8
Find the vertex, focus, and directrix of the parabola x²+4x+6y-2=0
(*Ans*) Center: (-2,1) Foci: (-2,-1/2) Directrix: y=5/2
Find the center, foci, and vertices of the ellipse (x-2)²/25+(y+1)²/9=1
(*Ans*) Center: (2,-1) Foci: (6,-1), (-2,-1 Vertices: (7,-1), (-3,-1)
What is the center and radius of the circle (x-5)²+(y+3)²=121
(*Ans*) Center: (5,-3) Radius: 11
If y varies *directly* as x, find the constant of variation k and write the equation when y=4 and x=20
(*Ans*) y=1/5x
What is the domain and range of the logarithmic function g(x)=lnx?
(*Ans*) D: all positive real numbers R: all real numbers
What is the domain and range of the exponential function f(x)=3^x?
(*Ans*) D: all real numbers R: all positive real numbers
If any, identify the horizontal, vertical or slant asymptotes. y=(x-3)/(x²-9)
(*Ans*) Vertical: x=-3 Horizontal: y=0
If any, identify the horizontal, vertical or slant asymptotes. y=(x-2)/(x²-25)
(*Ans*) Vertical: x=-5, 5 Horizontal: y=0
If any, identify the horizontal, vertical or slant asymptotes. y=(4x²-2x+1)/(x-1)
(*Ans*) Vertical: x=1 Horizontal/slant: y=4x+2
A deposit of $2,000 is made in a bank, which gives interest of 3% per year. a) What is the balance in the bank after 10 years if interest is calculate 12 times a year? b) What is the balance in the bank after 10 years if interest is given continuously?
(*Ans*) a) $2,698.71 b) $2,699.72
a) If f(x)=|x-8|, find f(-11) b) If f(x)=-(4-x)²+7, find f(-6)
(*Ans*) a) 19 b) -93
If y varies *inversely* as x, find the constant of variation k and write the equation when y=20 and x=9
(*Ans*) y=180/x
y varies *jointly as x and the cube of z*. If y=120 when x=5 and z=2, find the constant of variation k and its equation.
(*Ans*) y=3xz³
Find the polynomial function that has zeros 4,-3,3, and 0.
(*Ans*) y=x⁴-4x³-9x²+36
Find all the zeros of h(x)=x³+4x²+6x+4
(*Ans*) zeros: -2, -1+i, -1-i
For the polynomial y=x³-x²+6x-6, list all the possible rational zeros of the function.
(*Ans*) ±1
Write the quotient (3-8i)/(2+3i) in standard form.
(*Ans*) (18/13)-(25/13)i
Write log₃20 as a ratio of natural logarithms.
(*Ans*) (log20)/(log3)
If g(x)=(x²-9)/(9x), find g(n-2)
(*Ans*) (n²-4n-5)/(9n-18)
Write the equation of the ellipse in standard form and sketch it. 9x²+4y²+36x-24y+36=0
(*Ans*) (x+2)²/4+(y-3)²/9=1
Show that (x+3) and (x-2) are factors of the polynomial y=x⁴-x²+12x-36 and find the remaining factors of the polynomial.
(*Ans*) (x+3)(x-2)(x²-x+6)
Find the standard form of the ellipse with foci (0,0), (0,8) and major axis of length 12.
(*Ans*) (x-0)²/36+(y-4)²/20=1
Find the standard form of the equation of the parabola with vertex (1,2) and directrix y=1
(*Ans*) (x-1)²=4(3)(y-2)
Find the standard form of the ellipse with center (0,4), a=2c, vertices (-4,4) and (4,4)
(*Ans*) (x²)/(16)+(y-4)²/(16)=1
Find the standard form of the equation of the hyperbola with asymptotes y=±1/4 and vertices (±4,0)
(*Ans*) (x²)/16-(y²)/1=1
If f(x)=x²-4 and g(x)=√3-x, find (f/g)(x)
(*Ans*) (x²-4)/√3-4
Find the standard equation of the parabola with vertex (-1,2) and focus (2,2)
(*Ans*) (y-2)²=4(3)(x+1)
Find the standard form of the equation of the hyperbola with asymptotes y=±2x, and vertices (0,±2)
(*Ans*) (y²)/4-(x²)/1=1
If f(x)=x²-4 and g(x)=√3-x, find (f o g)(x)
(*Ans*) -1-x
Find the exact value of ln(1/√e)
(*Ans*) -1/2
Write 9i³-13i⁵ in standard form.
(*Ans*) -22i
Solve the inequality. (x+2)/(x-4)<0
(*Ans*) -2<x<4
Write √-3√-12 in standard form.
(*Ans*) -6
Solve the inequality. x³+2x²-4x-8≤0
(*Ans*) -∞≤x≤2
Find the exact value of 3lne⁴
(*Ans*) 12
Find the distance between points P(-3,2) and Q(2,-10)
(*Ans*) 13
Use properties of logarithms to simplify 7^(log₇15)
(*Ans*) 15
Find the exact value of log₅75-log₅3
(*Ans*) 2
Find the value of log₃20 to three decimals.
(*Ans*) 2.726
If f(x)=x²+√-4x, find f(-4).
(*Ans*) 20
Write the product in standard form. (3-2i)(5+4i)
(*Ans*) 23+2i
If f(x)=x²-4 and g(x)=√3-x, find (f-g)(-6)
(*Ans*) 29
Find the exact value of log₃27
(*Ans*) 3
Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of the polynomial y=x⁴-2x³+x-1.
(*Ans*) 3,1 or 0 positive zeros 1 or 0 negative zeros
Write the exponential equation 9^(3/2)=27 in logarithmic function.
(*Ans*) 3/2=log₉27
On a college campus of 5000 students, one student returns from vacation with a contagious and long-lasting virus. The spread of the virus is modeled by the formula y= 5000/(1+49999e^(-0.8t), t≥0 where y is the total number of students infected after t days. a) How many students are infected after 5 days? b) The college will cancel classes when 40% of students (2,000) are infected. After how many days will the college cancel classes?
(*Ans*) a) 54 students b) 10 days
If f(x)= { x²-2, x≤3 2x²+5, x>3 find a) f(-3) b) f(4)
(*Ans*) a) 7 b) 37
State the domain of a) y=(3x)/(3x-9) b) y= (x+3)/(x+2)(x-7)
(*Ans*) a) D: all real numbers except 3 b) D: all real numbers except -2 and 7
A keyboarder terns to type W words per minute after t weeks of practice, where W is given by W=100(1-e^(-0.3t). a) Find W(1) and W(8) b) After how many weeks will the keyboarder's speed be 95 words per minute?
(*Ans*) a) W(1)= 26 words/min, W(8)= 91 words/min b) 10 weeks
What is the domain of a) f(x) = (-10)/(x²-3x) b) g(x) = √(16-x²)
(*Ans*) a) all real numbers except 0 and 3 b) -4≤x≤4
Determine the a) center, b) vertices, c) foci, and d) slopes of the asymptotes of the hyperbola 25x²-9y²-100x-72y=269.
(*Ans*) a) center: (2,-4) b) vertices: (5,-4), (-1,-4) c) foci: (2+√34,-4), (2-√34,-4) d) ±5/3
Classify the graph of each equation. a) 3x²-2y²+4y-3=0 b) 2y²-3x+2=0 c) x²+4y²-2x-3=0 d) x²-2x+4y-1=0 e) 4x²-24x+4y²+16y-91=0
(*Ans*) a) hyperbola b) parabola c) ellipse d) parabola e) circle
The formula P=347e^kt gives the population of Reno, Nevada in thousands from 2000 through 2007, where t=0 corresponds to 2000. In 2005, the population of Reno was about 395,000. a) Find the value of k. b) Use the model to find the population of Reno in 2016. c) According to this model, during what year will the population of Reno reach 500,000?
(*Ans*) a) k=0.02591 b) P=535.255 c) 2014
a) Is f(x)=x⁵-3 even, odd, or neither? b) Is f(x)=x-x⁵ even, odd or neither?
(*Ans*) a) neither b) odd
Describe each transformation of y=f(x). a) y=3f(x) b) y=f(3x) c) y=f(x-4)+2 d) y=-f(x+4)
(*Ans*) a) vertical stretch by a factor of 3 b) horizontal compression by a factor of 3 c) horizontal shift 4 units to the right and vertical shift 2 units up d) reflection in the y-axis and horizontal shift 4 units to the left
Write y=x³-x²+6x-6 as a product of linear and quadratic factors.
(*Ans*) f(x)=(x-1)(x²+6)
Check the y-axis symmetry for y=x²-9
(*Ans*) has y-axis symmetry
Condense lnx-2lny+(1/2)lnz
(*Ans*) ln(xz^½/y²)
Write ln(3x²/5y³) in expanded form.
(*Ans*) ln3+2lnx-ln5-3lny
Write log4x²y³ in expanded form.
(*Ans*) log4+2logx+3logy
Condense log₅8-log₅t+log₅y
(*Ans*) log₅(8y/t)
Use long division to divide x⁶+5 by x²-1. Identify the quotient and the remainder.
(*Ans*) q(x)= x⁴+x²+1 r(x)=6
Describe the left-hand and right-hand behavior of the polynomial function f(x)=-2x³-10x²+14x+2
(*Ans*) the graph rises to the left and falls to the right
(NO CALCULATOR.) Sketch the graph of y=(5x-15)/(x²+x-6)
(*Ans*) x-int: x=3 y-int: y=5/2 zeros: asymptotes: x=-3,2; y=0
If any, find all the zeros of the rational function. y=(5/(x-2))+2
(*Ans*) x= -1/2
Find the x- and y-intercepts of y=(x+5)²-16
(*Ans*) x=-1,-9 y=9
Solve for x. e^2x=e^(x²-8)
(*Ans*) x=-2,4
Solve for x. 2^x=3^(x+1)
(*Ans*) x=-2.7095
Find all the zeros of y= x³-8x²+16x
(*Ans*) x=0, 4 (double zero)
Solve for x to 3 decimals. e^4=4
(*Ans*) x=1.386
Solve for x. log₂(x+3)=10
(*Ans*) x=1021
If any, find all the zeros of the rational function. y=(x²-4)/(x+2)
(*Ans*) x=2
Solve for x. log₂(2x-3)=log₂(x+4)
(*Ans*) x=7
Use 1-to-1 property to solve the equation for x. 2^(x-3)=16
(*Ans*) x=7
If x=2i is a zero of the polynomial y=x³-7x²+4x-28, find all the zeros of the polynomial.
(*Ans*) x=±2i, 7
Use 1-to-1 property solve the equation for x. log₇(x²-9)=log₇16
(*Ans*) x=±5
If f(x)=x²-4 and g(x)=√3-x, find (f-g)(x)
(*Ans*) x²-4-√3-4
Solve the inequality. x²-6x≥7
(*Ans*) x≤-1 or x≥7
Find the *point-slope form* of the equation of the line through (4,7), which is perpendicular to the line 3x-5y=-4
(*Ans*) y-7=(-5/3)(x-4)
Find the *slope-intercept form* of the line with points A(-3,5) and B(1,-5)
(*Ans*) y=(-5/2)x-(5/2)
Find the standard form of a parabola with vertex at the origin, a vertical axis, and passes through (4,6)
(*Ans*) y=(3/8)x²
Write the equation of the parabola f(x)=x²+8x+1 in *standard form.* Identify the vertex and axis of symmetry.
(*Ans*) y=(x+4)²-15 vertex: (-4,15) A.O.S: x=-4
Write the equation of the parabola whose vertex is (2,3) and passes through (0,2) in standard form.
(*Ans*) y=-1/4(x+2)²+3
The weight (w) of a synthetic ball varies directly with the cube of its radius (r). A ball with a radius 2 inches weighs 1.20 lbs. Find the equation for the weight of a ball of the same material with a 3-inch radius.
(*Ans*) y=0.15r³