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0 a Gt1qt 191 Btap 01 Sv Function 01

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  Calculus 1 (MT 1004) 191 – Exercises 01 1/ Find the domain and the range of ( ) 2 4  x x x f   −=  2/ For pictures 7 – 10, determine whether the curve is the graph of a function of x. If it is, state the domain and range of function. 3/ a/ Find the equation of the line segment joinings the points (1, –3) and (5, 7). Is this equation nonlinear?  b/ Find the equation of the top half of the circle ( ) 42 22 =−+  y x  4/ Find expressions for the quadratic functions whose graph are shown: 5/ Find an expression for a cubic function f: ( ) 61  =  f  &  ( ) ( ) ( ) 0201  ===−  f  f  f   6/ Three runners compete in a 100 – meter race. The graph describes the distance run as a function of time for each runners. Describe in words what the graph tell you about this race. Who won the race? Did each runner finish the race? 7/ The manager of s furniture factory finds that it costs 2200k to manufacture 100 chairs in one day and 4800k to produce 200 chairs in one day. a/ Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph.  b/ What is the slope of the graph and what does it represent? c/ What is the y – intercept of the graph and what does it represent? 8/ Find the domain of each function: a/ ( ) 22 1 11  x x ee x f  − −−=  b/ ( )  t  t g 21 −=  9/ Sketch the graph: a/ ( )  t t t  f  6 2 −=  b/ ( ) 5 −=  x xg  c/ ( ) ⎪⎩⎪⎨⎧>−≤≤−− −<+= 3,633,2 3,9  x x x x x x f   Topic: Exponent (Stewart , Section 1.5).  10/ (Exercises 24) Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? a/ One million dollars at the end of the month.  b/ One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, 1 2  − n  cents on the n th day, Topic: Inverse and logarithmic (Stewart , Section 1.6). 11/ A function is given by a table of values of a graph. Is it one – to – one? 12/ (Exercises 9 →  12) A function is given by a formula. Is it one – to – one? a/ f(x) =  x x 2 2 −   b/ ( )  x x f  310 −=  c/ g(x) =  x 1  d/ g(x) =  x cos  13/ (Exercises 21 →  16) Find a formula for the inverse of the function a/ ( )  x x f  310  −=   b/ ( ) 3214 +−=  x x x f   c/ ( )  x x ee x f  21 +=  d/ ( )  x x x x f  126 23 +−=  14/ (Exercises 61) Consider a bacteria population starts with 100 bacteria and doubles every three hours. Let n = f(t) be the number of bacteria after t hours. a/ Find f(t)  b/ Find the inverse of this function and explain its meaning. c/ When will the population reach 50000? 15/ (Exercises 62) When a camera flash goes off, yhe batteries immediately begin to recharge the flash’s capacitor, which stores electric charge given by ( ) ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ −=  − at  eQt Q 1 0 . (The aximum charge capacity is Q 0  and t is measured in seconds). a/ Find the inverse of this fnction and explain its meaning.  b/ How long does it take to recharge the capacitor to 90% of capacity if a = 2. 16/ (Exercises 69) Prove that ( ) 21 1sincos  x x  −= −  16/ (Exercises 70 →  72) Simplify the expression a/ ( )  x 1 sintan  −  b/ ( )  x 1 tansin  −  c/ ( )  x 1 tan2cos  −  Topic 4: Hyperbolic functions (Stewart , Section 3.11) 17/ (Exercises 1 →  6) Find the numerical value of each expression a/ 0sinh b/ 0cosh c/ 0tanh d/ 1tanh e/ ( ) 2lnsinh f/ 3cosh g/ 1cosh 1 −  h/ 1sinh 1 −  18/ (Exercises 11) Prove the identity: ( )  y x y x y x sinhcoshcoshsinhsinh  +=+  19/ According to the next identity ( )  y y y x y x sinsincoscoscos  −=+ , let guess the similar identity for hyperbolic functions. Write it down and prove it. 20/ (Exercises 20) If 1312tanh  =  x , find the value of the other hyperbolic functions at x. 21/ (Exercises 21) If 35cosh  =  x  and 0 >  x , find the value of the other hyperbolic functions at x. 22/ Show that ( )  x x f  sinh =  is one – to – one. Find explicitly formula for 1 −  f  . ------------------------------------------------------------------------------------------------------------
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