MATH 111 Departmental Midterm Exam Review Exam date: Tuesday, March 1 st. Exam will cover sections and will be NON-CALCULATOR EXAM.

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1 MATH Departmental Midterm Eam Review Eam date: Tuesday, March st Eam will cover sections and will be NON-CALCULATOR EXAM Terms to know: quadratic function, ais of symmetry, verte, minimum/maimum value, increasing/decreasing piece-wise defined function, continuous and discontinuous function, absolute value function, arithmetic combinations of function, function composition, inverse function, symmetry, function transformations, domain and range of the function, difference quotient, derivative of the function, polynomial function, degree, leading coefficient, zeros, roots, factors, complete factorization, long division, synthetic division, comple number, real and imaginary parts, comple conjugate, Be able to use the completion of squares to rewrite quadratic function in the verte form f ( ) = a( h) + k Be able to find the - and y-intercepts, ais of symmetry, verte, minimum/maimum value, increasing/decreasing intervals, domain and range and sketch the graph of the quadratic function Be able to use the quadratic formula to factor and solve the quadratic equation Know how to use the discriminant to determine the number of real solutions of the given quadratic equation 6 Know and be able to use vertical and horizontal shifts, vertical compression and elongation of the graph and symmetry to graph quadratic functions 7 Know the applications of quadratic functions 8 Be able to graph piece-wise defined functions and absolute value functions 9 Know the arithmetic combinations of functions, be able to find them and give the domain for each combination 0 Be able to find composition of two and three functions and give the domain for each composition Be able to write the function as the composition of few functions Be able to find the inverse function Be able to find the domain and range of the inverse function f ( + h) f ( ) Be able to find and simplify difference quotient h Be able to use the definition of the derivative to find the derivative of a quadratic, cubic, rational and square root functions 6 Be able to write an equation of the function in terms of one variable based on the given information 7 Be able to build a function from words 8 Be able to find the equation of the polynomial function from the given graph, and from the given conditions (zeros, intercepts, points, end behavior) 9 Be able to sketch the graph of the polynomial function based on the given conditions (zeros, multiplicity, end behavior, etc) 0 Know the Division Algorithm Be able to divide polynomial by another polynomial using long division and synthetic division Theorem to know: Intermediate Value Theorem, Factor Theorem, Remainder Theorem, Fundamental Theorem of Algebra, Complete Factorization Theorem, Conjugate Zeros Theorem, Rational Zeros Theorem Be able to find all eact zeros (rational, irrational and comple) of a function both algebraically and graphically Be able to recognize when a function has comple roots both algebraically and graphically, and find all eact comple roots of the polynomial

2 Be able to give complete factorization of the polynomial function of degree, and 6 Be able to verify if the indicated comple number is a zero of the polynomial function and use Conjugate Zeros Theorem to find ALL zeros of the polynomial function Partial Review Eercises Write the function in the form f ( ) = a( h) + k : a) f( ) = + ; b) y = + + ; c) f ( ) = Given f ( ) = and g( ) = Find a) f g; b) g f and give the domain for each composition Solve the given equation: a) + 8 = 0 ; b) = 0 If a projectile is shot vertically upward from the ground with an initial velocity of 00 feet per second, neglecting air resistance, its height s (in feet) above the ground t seconds after projection can be modeled by s = 6t + 00t a) How long will it take for the projectile to return to the ground? b) When will it reach maimum height? c) What is the maimum height? Sketch the graph, 0, < a) f ( ) = b) f ( ) = c) y = + 6, < +, > 6 An open rectangular bo with a volume of ft has a square base Epress the surface area A of the bo as a function of the length of the side of the base 7 Find the number of units that produce the maimum revenue, revenue in dollars and is the number of units sold R =, where R is the total Given f ( ) = 7, g( ) =, h( ) = + 6 Find f g h 9 Given f()= + ; g()= Find a) f g; b) g f and give the domain for each composition 0 Given f() = ; g() = - Find a) (f g)(); b) (g f)(7); c) (f f)() Given f() = and g() = f Find a) ( g ) (); b) (fg)() and give the domain for each combination f ( + h) f ( ) Find for the following functions: a) f() = 7 ; b) h c) f ( ) ; d) f ( ) = + f ( ) = + ; = A rectangle has perimeter 60 meters Epress the area A of the rectangle as a function of the length of one of its sides

3 A rectangle has an area of 6 square meters Epress the perimeter P of the rectangle as a function of the length of one of its sides A farmer wants to enclose a rectangular field by a fence and divide it into two smaller rectangular fields by constructing another fence parallel to one side of the field He has 000 yards of fencing Find the dimensions of the field so that the total area is a maimum 6 Epress the area A of a circle as a function of its circumference C 7 Find the derivative of a) f ( ) = 7 ; b) f ( ) = + 8 An open bo is to be made from a square piece of material centimeters on a side by cutting equal squares ( centimeters) from the corners and turning up the sides Determine the volume of the bo as the function of 9 Find the inverse function + a) f ( ) = ; b) f() = ; c) y = + ; d) y = ( ), ; e) f ( ) = + 0 A closed rectangular bo with a square base is and height h to be constructed from 00 square inches of material Epress the volume of the bo as a function of the length of the side of the base Use the table below to find the following values f() g() Find a) (fg)(); b) (gf)(); c) (f + g) () Find the maimum or the minimum value of f ( ) = 0 + Find the largest interval on which f ( ) = ( + ) is increasing Epress the distance d, from a point (, y) on the graph of + y = to the point (, 8) as a function of Determine the equation of the quadratic function whose graph is shown below An arrow is shot vertically upward with an initial velocity of 6 ft/s from a point 6 ft above the ground The height of an arrow above ground is described by s ( t) = 6t + 6t + 6 a) What is the maimum height attained by the arrow?

4 b) At what time does the arrow fall back to the 6-ft level? 7 A rectangular plot of land will be fenced into three equal portions by two dividing fences parallel to two sides If the total fence to be used is 00 m, find the dimensions of the land that has the greatest area 8 Find an equation of the polynomial function below that passes through (,-6) 9 A rectangular plot of land will be fenced into two equal portions by one dividing fence parallel to one side If the area to be enclosed is,000 m, epress the amount of fence required as a function of the width of the rectangular plot 0 a) If = and = are zeros of f ( ) = + 8 8, find all zeros of the polynomial b) If = i is a zero of f ( ) = + 8, find all zeros of the polynomial c) If = is a zero of f ( ) = + +, find all zeros of the polynomial d) If = is a zero of ( ) f = + +, find all zeros of the polynomial Use the Remainder Theorem to find r when f ( ) = + is divided by If = / is a zero of the polynomial f ( ) = , find the complete factorization of f() 8 i Write each epression in standard comple form: a + bi a) ; b) i Find a polynomial of lowest degree having only real coefficients and zeros at and + i Write your answer in epended form Find a polynomial of lowest degree having only real coefficients and zeros at = +i and = of multiplicity Write your answer in epended form 6 Determine the end behavior of P ( ) = ; y a as and y b as 7 If = i is a zero of the polynomial f ( ) = , find all zeros 8 Sketch the graph of polynomial P() that has zeros of multiplicity one at = 0 and =, has a zero of multiplicity three at = -, and satisfies P() as and P() as 9 Graph a) f ( ) = 8 ; b) f ( ) = + + ; c) p( ) = 6

5 Answers: a) ( ) 7 ; b) ( ) + ; c) ( ) + 9 a) 6, D (,] ; b), D [ 0, ) a) = ± 7 ; b) = / ; = 9 / a) 6 s, b) /8 s; c) 6/ ft a) b) c) A= + 8, > units 8 ( + 6) 7 9 a) ; D: (,] ; b) ; D: (, ) [/, ) + + ; 0 a) ; b) 6; c) 6 a) ; D:[0, ) b) ; D: [0, ] (a) 7 ; (b) + h +; (c) ; (d) ( + h )( ) + h A() = (0 - ) P() = + 7/ 00 by 70 6 A = 7 a ) 6 7; b) ( + ) 8 V() = ( ) ; + 9 a) f ( ) = ; b) + ; c) ( ) +, ; d) + ; e) f ) = 0 V = 7 a) 0, b), c) / (, ] (

6 d = + 0 y = ( ) 6 a) 70ft; b) s 7 0 m by 00 m 8 y = + 9,000 9 P = + 0 a) = -, =, = ± i ; b) = ± ; = ± i ; c) = ; = ±, d) =, = ± i ; e) =, = ± 7 ( )( )( + ) a) -i; b) i a =, b = = i 7 ; = ± 8 9 a) b) c) 9 0