Your exam contains 5 problems. The entire exam is worth 70 points. Your exam should contain 6 pages; please make sure you have a complete exam.

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1 MATH 124 (PEZZOLI) WINTER 2017 MIDTERM #2 NAME TA:. Section: Instructions: Your exam contains 5 problems. The entire exam is worth 70 points. Your exam should contain 6 pages; please make sure you have a complete exam. Box in your final answer when appropriate. Unless stated otherwise, you MUST show work for credit. No credit for answers only. If in doubt, ask for clarification. Your work needs to be neat and legible. You are allowed one sheet of notes (both sides). The only calculator allowed is TI-30x IIS Problem #1 (15 pts) Problem #2 (15 pts) Problem #3 (15 pts) Problem #4 (10 pts) Problem #5 (15 pts) TOTAL (70 pts) 1 1/12 Mid22Win17Sol (#46)

2 Problem 1 Compute the derivatives of the following functions. You do not need to simplify your final answer. a) (5 points) Let g(x) =sin( x 3 ), computeg (1). ( x x b) (5 points) f(x) = 3) 5 +x 2 1 c) (5 points) h(x) =ln(1+ln(1+ x)). 2 2/12 Mid22Win17Sol (2/6)

3 Problem 2 (15 points) a) A round solar panel is to be constructed with radius 5 meters, but errors in the construction are inevitable.use differentials to find a number δ with the property that, if the radius of the panel is within δ meters of the specified size, the area of the panel will be within 0.5 square meters of the desired area of 25 π meters. b) The average daily power output of a round solar panel of radius r in your area is given by f(r) =265 e x 5 2 watts. You do not have a calculator available to calculate exponentials so you want to use the tangent line approximation to estimate the output if you increase the radius to 6 m. Approximately, what is f(6)? c) Do you expect your estimate in b) to be an underestimate or an overestimatefor the true value of f(6)? Justify your answer without using a calculator to evaluate f(6) 3 3/12 Mid22Win17Sol (3/6)

4 Problem 3 : Consider the curve C of equation xy 2 + yx 2 =10 (a) (5 points) Find y (x) by implicit differentiation. (b) (5 points) Find the coordinates of all points P on C where the tangent is horizontal. (c) (5 points) Find the equation(s) of the line(s) tangent to C at P,forallthepoints P you have found in part b) 4 4/12 Mid22Win17Sol (4/6)

5 Problem 4 (10 points) Police are 30 feet from the side of the road. Their radar sees your car approaching at 70 feet per second when your car is 50 feet away from the radar gun, in other words your distance from the police car is changing at 70 feetpersecondwhenyou are 50 feet away from the police. The speed limit is 65 miles per hour,whichcorresponds to 95 feet per second. Are you speeding? 5 5/12 Mid22Win17Sol (5/6)

6 Problem 5 (15 points) An object is moving along the curve below withparametricequationsx = x(t), y = y(t), thecurveistangenttothelinepqinthepictureatthepointp(2,1). a) When the object reaches P its vertical speed is -3 feet/ min,whatisthehorizontal speed dx dt? b) Draw an arrow on the graph above to indicate in which direction the object is moving. Justify your answer. c) Find the tangents to the curve x(t) =t 2 sin(t), y(t) =2t cos(t)+2,t>0 i) at t= π 2. ii) at P=( 9π2 4,2). 6 6/12 Mid22Win17Sol (6/6)

7 Math 124 (Pezzoli) Winter 2017 Midterm #2 Name TA:. Section: Instructions: Your exam contains 5 problems. The entire exam is worth 70 points. Your exam should contain 6 pages; please make sure you have a complete exam. Box in your final answer when appropriate. Unless stated otherwise, you MUST show work for credit. No credit for answers only. If in doubt, ask for clarification. Your work needs to be neat and legible. You are allowed one sheet of notes (both sides). The only calculator allowed is TI-30x IIS Problem #1 (15 pts) Problem #2 (15 pts) Problem #3 (15 pts) Problem #4 (10 pts) Problem #5 (15 pts) TOTAL (70 pts) 1 7/12 Mid21Win17Sol (#31)

8 Problem 1 Compute the derivatives of the following functions. You do not need to simplify your final answer. ( x x a) (5 points) h(x) = 2) 3 +x 1 b) (5 points) g(x) = 1+ 1+ln(x). c) (5 points) Let f(x) =sin( x 3 ) compute f (2). 2 8/12 Mid21Win17Sol (2/6)

9 Problem 2 (15 points) a) A round solar panel is to be constructed with radius 10 meters, but errors in the construction are inevitable.use differentials to find a number δ with the property that, if the radius of the panel is within δ meters of the specified size, the area of the panel will be within 2 square meters of the desired area of 100 π meters. b) The average daily power output of a round solar panel of radius r in your area is given by f(r) =265 e x 10 5 watts. You do not have a calculator available to calculate exponentials so you want to use the tangent line approximation to estimate the output if you increase the radius to 11 m. Approximately, what is f(11)? c) Do you expect your estimate in b) to be an underestimate or an overestimatefor the true value of f(11)? Justify your answer without using a calculator to evaluate f(11) 3 9/12 Mid21Win17Sol (3/6)

10 Problem 3 : Consider the curve C of equation xy 2 + yx 2 =2 (a) (5 points) Find y (x) by implicit differentiation. (b) (5 points) Find the coordinates of all points P on C where the tangent is horizontal. (c) (5 points) Find the equation(s) of the line(s) tangent to C at P,forallthepoints P you have found in part b) 4 10/12 Mid21Win17Sol (4/6)

11 Problem 4 (10 points) Police are 30 feet from the side of the road. Their radar sees your car approaching at 80 feet per second when your car is 50 feet away from the radar gun, in other words your distance from the police car is changing at 80 feetpersecondwhenyou are 50 feet away from the police. The speed limit is 65 miles per hour,whichcorresponds to 95 feet per second. Are you speeding? 5 11/12 Mid21Win17Sol (5/6)

12 Problem 5 (15 points) An object is moving along the curve below withparametricequations x = x(t), y = y(t), thecurveistangenttothelinepqinthepictureatthepointp (2,1). a) When the object reaches P its vertical velocity is 2 feet/ min, what is the horizontal velocity dx dt? b) Draw an arrow on the graph above to indicate in which direction the object is moving. Justify your answer. c) Find the tangents to the curve x(t) =t 2 sin(t), y(t) =3t cos(t)+1,t>0 i) at t= π 2. ii) at P=( 9π2 4,1). 6 12/12 Mid21Win17Sol (6/6)

. Section: Your exam contains 4 problems. The entire exam is worth 60 points.

Math 125 Section D (Pezzoli) Fall 2017 Midterm #1 (60 points) Name TA:. Section: Your exam contains 4 problems. The entire exam is worth 60 points. This exam is closed book. You may use one 8 1 11 sheet