Math 111 Final Exam Review

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1 Math 111 Final Eam Review With the eception of rounding irrational logarithmic epressions and problems that specif that a calculator should be used, ou should be prepared to do the entire problem without the use of a calculator. 1. Use the graph of = f() in Figure 1 to answer the following. Approimate where necessar. Evaluate f( 1). (b) Evaluate f(0). (c) Solve f() = 0. (d) Solve f() = 7. (e) Determine if f is even, odd, or neither from its graph. (f) State an local maimums or local minimums. (g) State the domain and range of f. (h) Over what interval(s) is the function increasing? (i) Over what interval(s) is the function decreasing? (j) Over what interval(s) is the function concave up? (k) Over what interval(s) is the function concave down? (l) Find the zeros of f. (m) Find a possible formula for this polnomial function. Figure Let f() = +. Find f 1 (). (b) Confirm the inverse b computing f 1 (f()) and f ( f 1 () ). (c) State the domain and range of f and f 1. (d) Evaluate f(0). (e) Solve f() = 3. (f) Algebraicall determine if f is even, odd, or neither. (g) State an horizontal and vertical asmptotes. (h) State an horizontal and vertical intercepts. (i) Sketch a graph of = f() in Figure. Figure 1

2 3. Let f() =. For each of the following, sketch a graph of the transformation in Figure and write the simplified formula for the function. Describe the order of transformations, being as specific as possible and listing them in an appropriate order. = f() (b) = f() (c) = f()+3 (d) = f(+1)+3 (e) = f(3) (f) = 3f( (+)) 1 Figure 3. Graph of = (b) (c) (d) (e) (f) Figure. For each function below, identif the original (or basic) function and eplain how the graph is a transformation of the graph of the original function. State all steps to this transformation in an appropriate order. g() = 1 3 (( 7))3 (b) g() = 7ln(+)+5 (c) g() = 1 (+1) 3 5. The point (,1) is on the graph of = f(). Determine the point on the graph of... = f( 5) 7 (b) = f() (c) = 3f( ) (d) = 1 8 f((+3))+5 Instructor: A.E.Car Math 111 Final Eam Review Page of

3 . Complete Table 1 below using the given values in the table. If an value is undefined, write undefined. Table f() g() (g f)() (g f)() f()+g() f() g() f 1 () 7. Find a formula for the piecewise-defined function graphed in Figure 5 below. f() = 8. In Figure, graph the piecewise function defined b, 3 < 0 f() =, 0 < < 1 1 +, 1 Figure 5. Graph of = f() Figure. Graph of = f() 9. The volume, V(r) (in cubic centimeters) of a circular balloon of radius r (in centimeters) is given b V(r) = 3 πr3. As someone blows air into the balloon, the radius of the balloon as a function of time t (in seconds) is given b r = g(t) = t. Find and interpret V(3). (b) Find and interpret g(3). (c) Find and interpret V(g(3)). (d) Find and interpret V(g(t)). (e) Eplain wh g(v(r)) in nonsense. Instructor: A.E.Car Math 111 Final Eam Review Page 3 of

4 10. Find the following for the functions f, g, and h defined b f() = 3+1 g() = 3 +1 h() = 5 f (g()) (b) (h f)(1) (c) (h+g)(1) (d) (g g)(0) (e) (f g)(0) (f) (g g)() (g) (f g)() (h) (g h)() (i) (h h)() 11. Find the algebraic rule (or formula) of an eponential function that passes through each pair of points: ( 1, 1 3) and (1,1) (b) (,18) and (5,) 1. Find the eact values of the epressions below. log () (b) ln( ( e) (c) log 1 ) Solve the following equations. Give the eact solution and then round accuratel to two decimal places. Clearl state each solution set. 7 1 = (b) e 5 = 10 (c) 5e = 10 (d) 3 = 9 + (e) 3 +1 = (f) 5 7 = 3 7 (g) log (+1) = (h) log ()+log (3) = log () (i) log () log (3) = log () (j) log 5 ( ) = log 5 () (k) log ( 3 ) = 1 (l) log(1 ) = +log(1+) (m) log (+)+log (+3) = 1 1. The percentage of carbon 1, Q, remaining in a fossil t ears since deca began can be modeled b the function Q = f(t) = 100e t If a piece of cloth is thought to be 750 ears old. What percentage of carbon 1 is epected to remain in this sample? (b) If a fossilized leaf contains 70% of its original carbon 1, how old is the fossil? 15. The temperature of a cup of tea after it was brewed can be modeled b the function T(t) = 100e 0.1t +8, where t is the number of minutes since the tea was brewed and T(t) is the temperature in degrees Fahrenheit at time t. Find and interpret T(0). (b) Find and interpret T(10). (c) Solve and interpret T(t) = 80. (d) Graph = T(t) in our calculator. What is the horizontal asmptote? Instructor: A.E.Car Math 111 Final Eam Review Page of

5 1. Tom and Jerr make separate investments at the same time. Their respective investments can be modeled b the functions ( T(t) = 5000(1.05) t and J(t) = ) 1t 1 where t is the number of ears since each investment began and T(t) and J(t) are their respective investment values in dollars. How much does Tom invest initiall? How much does Jerr invest initiall? (b) What will the values of their respective investments be after 5 ears? (c) How long will it take for Jerr s investment to double? 17. The acidit of a solution is measured using the ph scale. Luck for ou, it s a logarithmic scale! The ph depends on the concentration, or molarit, in moles/liter of hdrogen ions. The formula for ph is given b ph = log ([ H +]), where [H + ] is the molarit of hdrogen ions. A low ph is ver acidic (like lemons) and a high ph is ver basic (like bleach). Coffee has a hdrogen ion concentration of about [ H +] = Find the ph of coffee. (b) Pure water is called neutral and has a ph of 7. Find the hdrogen ion concentration of pure water. (c) The hdrogen ion concentration of bleach is about [ H +] = Find the ph of bleach. (d) The ph of pure Hdrochloric acid (HCl) is 0. Find the hdrogen ion concentration of HCl. 18. Let g() = Use our calculator to find the zeros of the function. (b) Use our calculator to estimate the interval where the graph is concave down. Round accuratel to the nearest tenth. (c) Use our calculator to estimate the local maimum value and where it occurs. (d) Use our calculator to estimate the absolute minimum value and where it occurs. 19. Find possible formulas for each polnomial function in Figure 7. Clearl list the zeros and their multiplicities. (1, ) - (b) Figure 7 Instructor: A.E.Car Math 111 Final Eam Review Page 5 of

6 0. Sketch a graph of = f() for each polnomial function below. Also list the zeros and their multiplicities, the vertical intercept, and the long-run behavior. f() = (+) (b) g() = ( )(+1) (+) 1. Find possible formulas for each rational function in Figure 8 below. List the zeros and their multiplicities, an vertical asmptotes and an horizontal asmptotes. = =0 (1, ) =1 =-1 = (b) Figure 8. Find a possible formula for the rational function in Figure 9 below. List the zeros and their multiplicities, an vertical asmptotes and an horizontal asmptotes. 10 =-3 =3 = 8 = Figure 9 Instructor: A.E.Car Math 111 Final Eam Review Page of

Math 111 Final Exam Review KEY

Math 111 Final Exam Review KEY Math 111 Final Eam Review KEY 1. Use the graph of = f in Figure 1 to answer the following. Approimate where necessar. a b Evaluate f 1. f 1 = 0 Evaluate f0. f0 = 6 c Solve f = 0. =, = 1, =, or = 3 Solution