Math 140 Review for Quiz 13 page 1

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1 Math Review for Quiz age. Solve each of the following sstems of equations over the real numbers. a) + x + x + = x = x + = x d) x + = x =. Solve each of the following equations. a) x x + = x + x = x + x = d) x x + =. Solve each of the following equations. a) log (x + ) = log (x + ). Solve each of the following equations. = ln (x + ) = d) ln (x ) = a) x = x = e x+ = e d) e x+ =. Find the domain for each of the following functions. a) f (x) = x + g (x) = x + h (x) = log (x + ) d) f (x) = x x e) g (x) = x x f) h (x) = ln x x g) f (x) = x x + h) f (x) = x x i) f (x) = ln x + x. Solve each of the following inequalities. x + a) < x + x x d) x x + < e) x x +. a) Two sides of a triangle are a = inches and b = inches long. What can be said out the third side? Two sides of a right triangle are a = inches and b = inches long. What can be said out the third side?. a) Comute the exact value of the area of a triangle with sides m, m, and m. Comute the exact value of the area of an equilateral triangle if all three sides are m long.. Is there a right triangle with integer sides so that the di erence between the longest side and the middle side is and the di erence between the middle side and the shortest side is also?. Find the exact value of the length of the main diagonal in a rectangular rism with sides,, and units.. The sum of two numbers is. The sum of their recirocals is. Find these numbers.. a) The hotenuse of a right triangle is units. The di erence between the lengths of the other two sides is units. How long are the sides of this triangle? The shortest side of a right triangle is units. The di erence between the lengths of the other two sides is unit. How long are the sides of this triangle?. An arch is in the shae of a semicircle. At a oint along the base foot from an end of the arch, the height of the arch is feet. Find the maximum height of the arch.. Consider the oints A (; ) and B (; ). a) Comute the exact value of the distance between A and B. Write an equation for the line that asses through A and B.. Comute the exact value of the height in a ramid if its base is a square and all of its edges are meters long. c Hidegkuti,

2 Math Review for Quiz age. Simlif each of the following. a) a b a b a b a + b x +. Simlif each of the following. a) log P log Q log ( x ) d) log ( ) e) ln e P f) ln e g) e ln v u rq B Bts CC. AA h) log i) log j) log : d) k) log () a + b c d l) log ( ) m) log = n) log o) log ) e ln A. Simlif each of the following. a) (x + ) (x ) x (x ) a + b a b. Grah each of the following functions. Also, state the function s basic roerties. a) f (x) = x f (x) = log x f (x) = x. Re-write : = : :::: as a fraction of two integers.. Simlif each of the following. a) + e) + + d) + f). List the given numbers in an increasing order (from smallest to greatest) e log e log log. Suose that f and g are functions, de ned as f (x) = (x ) and g (x) = x. Comute each of the following. a) f () g () f (g ()) e) f (g ()) g) f (g (x)) f () + g () d) g (f ()) f) g (f ()) h) g (f (x)). We are standing on the to of a feet tall building and throw a small object uward. The object s distance, measured in feet, after t seconds is h (t) = t + t + a) How long will it take for the object to hit the ground? What is the highest oint that the object reaches?. Find an equation for the line that is a) assing through the oint (; ) and is arallel to x + =. assing through the oint (; ) and is erendicular to x + =. asses through (; ) and (; ). c Hidegkuti,

3 Math Review for Quiz age. Suose that and q are real numbers such that is three greater than twice q. Find each of the following. a) the smallest value of + q the smallest value of q the greatest value of q. Solve each of the following inequalities. a) x x < x x x + > x d) x + < x. A coman nds that if the rice their roduct at $, the can sell items of it. For ever dollar increase in the rice, the number of items sold will decrease b. Find the maximum revenue for the coman and also, the rice that guarantees the maximum revenue.. We have liters of a % sugar solution. How man liters of a % sugar solution needed to be added so that the resulting mixture is a % solution?. We have two di erent solutions. Of we mix gallons of the rst solution with gallons of the second solution, the resulting mixture is a % solution. If we mix gallons of the rst solution with gallons of the second solution, the resulting mixture is a % solution. Find the concentration of these two solutions.. We asked a man out his age. He resonded: "If ou added m age, and half of m age, and quarter of m age, and one ear, the sum is ears." How old is he?. In Paul s class, the ratio of bos to girls is to. Yesterda, bos and girls were sent due to a eld tri. Because of this, the ratio of bos resent to girls resent was to. How man girls are in Paul s class?. We needed a shorter roe. So we grbed a rett long roe, we rst cut o a quarter of its length, and then another meters. Now the roe is meters long. How long was it originall?. Which one is greater? (Hint: use algebra) a) or +. Which one is greater? or + a) or or +. One third of a roe is meters longer than one fth of the roe. How long is the roe?. Comute the following without calculator. (Hint: tr to use algebra.) a). Find all ositive numbers with the following roert: the number is one greater than its own recirocal.. We urchased some stocks. At rst, the value of the stocks increased b %: Then the value decreased b %. Exress the two changes as a single change. (Hint: it is not a % increase.) c Hidegkuti,

4 Math Review for Quiz age Answers. a) (; ) and ( ; ) (; ) no real solutions d) (; ) and (; ). a) ( doesn t work) ; d) ( doesn t work). a) e d) e +. a) ( + log ) d) ( + ln ). a) ( ; ) [ ( ; ) [ ; ) ( ; ) d) ( ; ) [ (; ) [ (; ) e) [; ] f) (; ) g) R h) [; ) [ (; ) i) ( ; ) [ ( ; ) [ (; ) [ (; ). a) ( ; ) [ ( ; ) ; [ (; ) ; d) ( ; ) e) ( ; ). a) in < c < in in or in. a) m m. ;, and. units. and. a) ; and units long ; ; and units long. ft. a) unit = unit. m = m. a) (x ) = + or (x ) = or = x + b a. a) P Q x d) e) P f) l) undened m) n) o) ).. a) x x + x x x + d) ad + b d cd g) h) i) j) k) A.. a) x x + x. a) f (x) = x f (x) = log x f (x) = x x - x x - domain: fx : x = g range: f : = g odd not continuous no max or min - domain: (; ) range: R not odd, not even continuous increasing no max or min domain: [; ) range: [; ) not odd, not even continuous increasing minimum: (; ).. a) d) e) f) c Hidegkuti,

5 Math Review for Quiz age. log < e < < e < log < log. a) d) e) f) g) x x + h) x x. a) seconds feet. a) = x. a). a) ; + = x = x ( ; ] [ [; ) x = d) no solution. a revenue of $, when the rice is $. liters. % and %... meters long. a) +. a) the are equal. : meters. a). +. :% increase Last revised: October, c Hidegkuti,