Math 1 Midterm Eam Review with Answers Name Date The Review has 16 questions. Simplify all answers, include all units when appropriate. 1. [Sec. 1.] Solve the following problems. a. A company s profit increased linearly from $6 million at the end of year 1 to $1 million at the end of year. i. Use the two data points (year, profit) coordinates as (1,6) and (,1), to find the linear relationship y m b between years and y profit. ii. Find the company s profit at the end of 5 years. iii. Predict the year when the profit will be 0 million. b. A Telephone company charges a client $.95 for a service of 00 minutes of long distance, and charges another client $.95 for service of 00 minutes of long distance calls. i. Use the data points (minutes, charge) coordinates to find the linear relationship y=m+b between number of minutes and y dollars charged. ii. Find the company s charge for 100 minutes of long distance service. iii. Find the number of minutes of service for a client that was charged $8.15.. [Sec. 1.] Given the function a. f ( ) b. f ( ) 9 i. Write the domain in set notation. ii. Find the and y intercepts if any, and write as order pair. iii. Evaluate f () iv. Graph the function, label three points. Page 1 of 11
Math 1 Midterm Eam Review with Answers. [Sec. 1.] a) A group of students want to raise money to attend an international conference; the students want to build canoes during the summer in an old warehouse. The rental of the warehouse is $1500 for the summer, and the materials needed to build one canoe cost $15. The canoes can be sold for $75 apiece. i. Write the equation for the Profit. ii. How many canoes must the students sell to break even? iii. How many canoes must the students sell to make a profit of at least $1000? b) A manufacturer can produce bookcases at a cost of $80 apiece. Sales figures indicate that if the bookcases are sold for dollars apiece, approimately150 bookcases will be sold each month. i. Epress the manufacturer s monthly profit as a function of selling price. ii. Find the price where the manufacturer breaks even. iii. Find the price to maimize the profit. Page of 11
Math 1 Midterm Eam Review with Answers. [Sec. 1.1] If, a) i. Find f ( ) and g( ) iv. ii. Determine the domain of the function and, write in set notation. v. iii. Find the composition f ( g( )), simplify if possible. iv. Find the composition g ( f ( )), simplify if possible. b. If and i. Find ii. Determine the domain of the function and, write in set notation. iii. Find the composition f ( g( )), simplify if possible. iv. Find the composition g ( f ( )), simplify if possible. 5. [Sec1.6] a. Graph f ( ) 8 Label four points. Page of 11
Math 1 Midterm Eam Review with Answers ( ) ( ) EXIST / NOT EXIST (Circle One) b. Graph f ( ) Label four points. 7 ( ) ( ) ( ) ( ) EXIST / NOT EXIST (Circle One) Page of 11
Math 1 Midterm Eam Review with Answers 6. [Sec. 1.5] Find the it or if it does not eist. If it is infinite indicate whether it is + or - a. Lim b. Lim 8 6 7 7 7 ( ) c. Lim 0 d. Lim 16 5 e. Lim 1 f. Lim 1 5 7. [Sec.1] For the given function use the definition of the derivative to find f '( ), Show all the required steps. a. f ( ) b. f ( ) 7 8. [Sec..] Find the derivative, f '( ) a. f ( ) 5 6 b. f ( ) c. f ( ) d. f ( ) 5 9. [Sec.] Find the derivative of the function using the product rule of derivatives, f (), simplify. a. f ( ) ( )( 1) b. f ( ) ( )( ) Page 5 of 11
Math 1 Midterm Eam Review with Answers 10. [Sec.] Find the derivative, y, simplify if possible. a. 1 y b. 1 y 1 5 11. [Sec.] Find the derivative, f (), simplify if possible. 1 a. f ( ) b. f ( ) 1. [Sec..] Evaluate the second derivative of the given function. a. f ''(1) for the function f ( ) 5 b. f ''(1) for the function f ( ) 1. [Sec.5] Solve the following applied problems, interpret results a. A steel mill finds that its cost function is C ( ) 8000 600 dollars, where is the daily production of steel (in tons). i. Find the Marginal Cost Function. ii. iii. Find the marginal cost when 6 tons of steel are produced. Interpret the results. b. The manufacturer determines that the profit P (in dollars) derived from selling units is given by P( ) 0.000 7. i. Find the Marginal Profit Function. ii. Find the marginal profit for a production level of 5 units. iii. Interpret the results. Page 6 of 11
Math 1 Midterm Eam Review with Answers 1. [Sec..] A company s profit function is P ( ) 1 1700 a. Find the Relative Rate of Change of the Profit Function. b. Find the Marginal Profit Function. c. Evaluate the Relative Rate of Change of the Profit Function at = 50. 15. [Sec..] Solve the following applied problems, interpret results. d. The number of weeds (in hundreds) n in lawn weeks after a weed killer was applied is given by n ( ). Find the rate of change in the number of weeds in the lawn at the end of the second week. e. The velocity of an object in meters per second is v ( t) 1 1 9t, 0 t. Find the acceleration of the subject when t =.15 sec. 16. [Sec.1 &.] For the following functions: f. The function, f ( ) 6 9 1, g. The function, f ( ) 0 1 i. Make a sign diagram of f ' ( ) ii. Identify the relative etreme points and inflection points iii. Sketch the graph Page 7 of 11
Math 1 Midterm Eam Review with Answers Midterm Review Math 1 Answers: #1 a. i. y = + ii. y = $ million iii. @ the end of the 7 th year b. i. y =.1 +.95 or y 1 10. 95 ii. y = $ 1.95 iii. minutes # a. i. ii. -intercept: None & y-intercept: iii. f () = iv. Graph curve with label on any of three points (, ); (1, -); ) b. i. { - } ii. -intercept: iii. f () = iv. Graph curve with label on any of three points: (0, -1); (, ); (-,) # a. i. P ( ) 150 1500 ii. 10 canoes iii. 17 canoes b. i. P ( ) 0 1000 ii. @ price of $80 and $150 per unit iii. price of $115 per unit # a. i. f(-) = -1; f(() = ; g() = 9 ii. Domain f() {R}; g() {R} iii. f(g()) = 9 iv. g(f()) = 9 b. i. f(-) = Undefined; f(0) = Undefined; g() = 5 1 ii. Domain f() ; ; g(){r} iii. f ( g( )) 6 iv. g ( f ( )) 1 1 Page 8 of 11
Math 1 Midterm Eam Review with Answers #5 a. Piecewise lines with points (0, 8), (1, 7), (, 6), (, 5); stop at the point (, 5) And line with points (, ), (, 5), (5, 7), (6, 9); (, 5) do not eist. f( ) 5 f( ) f( ) do not eist. b. Piecewise lines with points (0, 7), (1, 5), (, ), (, 1), (, -1); stop at the point (, -1) and line with points (, ), (5, ), (6, ); (, -1) do not eist. f( ) 1 f( ) f( ) do not eist. #6 a. 0 b. d. Does not eist e. 0 c. #7 a. f () = 8 b. f () = - #8 a. f () = 0 6 + 6 b. f () = 1 8 9 c. f () = 5 - d. f () = 10 5 #9 a. f () = 9 + 8 + 1 b. f () = 16 6 5 #10 a. y = ( 1) 10 b. y = ( 5) 7 Page 9 of 11
Math 1 Midterm Eam Review with Answers (1 ) #11 a. f () = ( ) b. f () = ( ) #1 a. f (1) = 5 b. f (1) = 9 #1 a. i. MC() = 000 00 ii. MC(6) = 87.5 iii. The cost is increasing by $87.5 for every ton of steel when the mill is producing 6 tons of steel. b. i. MP() =.001 + 7 ii. MP(5) = 10. iii. The profit is increasing at $10. per unit when manufactures produces 5 units. #1 a. RRCP() = 1 1 1700 b. MP () 1 c. RRCP(50) = 0.008 #15. a. n ()=-, the rate of change is decreasing hundred units per week. b. a(.15) = -.7, the object is decreasing in acceleration of.7 m/sec.. #16. a. i) + - + ii) Etreme Points (1,) Relative Ma (,-1) Relative Min. Inflection Point (, 1) b. i) - + - + ii) Etreme Points (-5,-87) Rel Minimum (0,-1) Rel. Maimum (, -) Rel. Minimum. Inflection points (1.0817, -8.97), (-.08, -8.8) Page 10 of 11
Math 1 Midterm Eam Review with Answers Page 11 of 11