M160 Final Exam, Fall 2006 page 1 Name: _______________________1. (24 points) Evaluate the following limits, including the possibility that the limit might not exist. Show details of how the limit can be found or shown not to exist using neither a calculator nor l’Hospital’s rule.(a) 1limxxx1(b)0limhhh 164(c) xlimxxx33(d)2limx22)2(2xxxM160 Final Exam, Fall 2006 page 2 Name: _______________________2. (24 points) Use differentiation formulas (not your calculator) to find the following derivatives. Show the details of your calculations. Simplify yo u r an swers . (Collect like terms and/or factored form. Ask if uncertain.)(a) f(t ) = 195822ttf′ (t) = ____________________________________(b) g( θ ) = (θ + 1)4 (θ4 + 1)g′ (θ) = _________________________________________________(c)xxdxd 1sin = ____________________________________(d) y = tan2 xdxdy = __________________________________and 22dxyd = _________________________________M160 Final Exam, Fall 2006 page 3 Name: _______________________3. (24 points) Evaluate the following indefinite and definite integrals by using the Fundamental Theorem and antidifferentiation formulas, NOT A CALCULATOR. Exact answers (not decimal approximations) are required for definite integrals. Simplify your answers.(a)10)15)(32( dxxx = (b)dxxxx )2(2132 =(c) dxxxx )tan(secsec =(d) 9132313dsss =M160 Final Exam, Fall 2006 page 4 Name: _______________________4. (16 points) (a) Consider the function g(x ) = 000,,10,1xxxxx. Without using a calculator, evaluate0limxg( x ) or explain how to see that this limit does not exist.(b) Explain accurately in non-technical language what it means to say that a function y = f(x) has limit L as x approaches a.(c) Albert says that the limit as x approaches 0 of the function g( x) defined above is L = 0. Use your explanation of what it means for a function to have a limit to explain why Albert is wrong.5. (16 points) (a) State the mathematical definition of the derivative of a function y = f(x ) at a point x = c. (c) Sketch the graph of the function g(x) =11,)2/1(,2/12xxxx. Decide by examining the graph whether this function differentiable at x = 1. State your conclusion and describe what you see in the graph that leads you to this conclusion.(d) Use the definition from (a) to show that your conclusion in (c) is correct.M160 Final Exam, Fall 2006 page 5 Name: _______________________6. (15 points) (a) Carla used her TI-89 calculator to conclude that dxxx123 = Cxx 2/122/32)1()1(31.Use what you know about indefinite integrals to confirm that Carla’s calculator is correct.(b) Evaluate the definite integral 30231dxxx without a calculator. The final answer involves square roots. (c) Write the name of the theorem you used to evaluate the integral in (b). ________________________________7. (15 points) (a) Sketch an accurate graph of the function f (x) = x34x.(b) Explain how to see immediately from the graph you sketched in (a) that dxxx 1143 = 0. (c) Make the change of variable (u-substitution) u = x2 in dxxx 1143, including changing the limits of integration. Show details.(d) Use the result of (c) to explain how to see w it h out u sin g a calcu lat or that dxxx 1143 = 0.M160 Final Exam, Fall 2006 page 6 Name: _______________________8. (16 points) (a)Sketch the graph of the function f(x) =32)31()4( xx on the coordinate system given. Describe what you see in the graph that tells you this function has three critical points. (b) Use calculus (not a calculator) to find the exact values of the critical points of f(x) = 32)31()4( xx. Show details. Explain how to tell from your calculations that each point you found is a critical point.(c) What is the sign of the second derivative of f(x) = 32)31()4( xx at the second largest critical point? Explain what you see in the graph that tells you.9. (16 points) (a) On the given coordinate system, draw the line y = 2x and the point (3, 1).(b) Use the figure from (a) to estimate the coordinates of the point on theline y = 2x closest to (3 ,1). Show how you got this estimate.(c) Use calculus methods to find the coordinates of the point on the line y = 2x closest to the point (3, 1).Use the first derivative test to verify that the point you found minimizes the distance between the line and the point.M160 Final Exam, Fall 2006 page 7 Name: _______________________10. (17 points) This problem involves finding the volume of the solid of revolution formed by revolving the region R enclosed by the x-axis and the graph of y = 4 x – 2x2 around the y-axis.(a) In the given figure, sketch this solid of revolution.(b) In the figure, sketch a representative volume element used to set up the integral for the volume.(c) Write a definite integral that represents the volume.(d) Find the exact volume of this solid of revolution (not a decimal approximation!).11. (17 points) Obviously the x-coordinate of the center of mass of a thin plate of uniform density  in the shape of the region R enclosed by the x-axis and the graph of y = 4 x – 2 x2 is x = 1. Find the y-coordinate of the center of mass of a thin plate of uniform density  in the shape of the region