More Differentiation Page 1 Directions: Solve the following problems using the available space for scratchwork. Indicate your answers on the front page. Do not spend too much time on any one problem. Note: Let ln(x) denote the natural logarithm of x with base e. 01. 07. 13. 19. 25. 31. 02. 08. 14. 20. 26. 32. 03. 09. 15. 21. 27. 33. 04. 10. 16. 22. 28. 34. 05. 11. 17. 23. 29. 35. 06. 12. 18. 24. 30. 36. 01 If ( ) then ( ) ( )( ) ( )( ) ( )( ) E) ( )( ) Page 1 of 20
More Differentiation Page 2 02 If ( ) then ( ) E) 03 The position of a particle on the x-axis at time t, t > 0, is ln (t ). The average velocity of the particle for is E) Page 2 of 20
More Differentiation Page 3 04 Let ( ) be a continuous and differentiable function on the interval and let ( ) ( ) The table below gives values of ( ) the derivative of ( ) What is the value of ( ) x 0.1 0.2 0.3 0.4 0.5 0.6 ( ) 1.010 1.041 1.096 1.179 1.298 1.486 1.010 1.096 1.486 3.030 E) 3.288 05 Let f and g are twice differentiable functions such that ( ) ( ) ( ) ( ) then ( ) ( ) and ( ) ( ) ( ) ( ( )) ( ( ) ( )) ( ( )) ( ) E) ( ) ( ) Page 3 of 20
More Differentiation Page 4 06 The figure above shows a road running in the shape of a parabola from the bottom of a hill at A to point B. At B it changes to a line and continues to C. The equation of the road is ( ) { B is 1000 feet horizontally from A and 100 feet higher. Since the road is smooth, ( ) is continuous. What is the value of b? 0.02 0.002 0.0002 E) 0.00002 Page 4 of 20
More Differentiation Page 5 07 Let ( ) and let g be the inverse of f. What is the value of ( ) E) 08 The function ( ) ( ) has one zero in the interval [ ]. The derivative at this point is E) i Page 5 of 20
More Differentiation Page 6 09 Let ( ) be a differentiable function. The table below gives the value of ( ) and ( ), the derivative of ( ), at several values of x. If ( ) ( ) what is the value of ( ) x 1 2 3 4 ( ) -3-8 -9 0 ( ) -5-4 3 16 E) Page 6 of 20
More Differentiation Page 7 10 What is the absolute maximum value of the function? E) -4 11 If ( ) find the value of c that satisfies the Mean Value Theorem on the closed interval [0, 2]. 2 E) Page 7 of 20
More Differentiation Page 8 12 Let ( ) be a differentiable function defined only on the interval [ 2,10]. The table below gives the values of ( ) and its derivative ( ) at several points in the domain. x -2 0 2 4 6 8 10 ( ) 26 27 26 23 18 11 2 ( ) 1 0-1 -2-3 -4-5 The tangent to the graph of ( ) and parallel to the segment between the endpoints intersects the y-axis at the point ( ) ( ) ( ) ( ) E) ( ) Page 8 of 20
More Differentiation Page 9 13 If then at the point ( ) is E) 14 If then at the point ( ) is E) Page 9 of 20
More Differentiation Page 10 15 If, amd what is the value of at 0 1 E) 16 If h y i ( ) ( ) E) None of the Above Page 10 of 20
More Differentiation Page 11 17 If h y i E) None of the Above 18 If ( ) ( ) ( ) then ( ) ( ) ( ) ( ) E) Page 11 of 20
More Differentiation Page 12 19 At what point on the curve is the tangent line vertical? ( ) only ( ) only ( ) only ( ) and ( ) E) The tangent line is never vertical 20 Let f be the function given by ( ) and let g be the function given by ( ). At what values of x in the interval do the graphs of f and g have parallel tangent lines? 0 E) 2.207 Page 12 of 20
More Differentiation Page 13 21 What is the equation of the line tangent to the graph of ( ) the point where ( ) at ) ) ) ) 22 At how many points on the curve curve perpendicular to the line is the normal to the 0 E) 4 Page 13 of 20
More Differentiation Page 14 23 The curves and intersect in the first quadrant. The angle at which they intersect is 30 0 45 0 60 0 90 0 E) 0 0 24 The line tangent to the graph of at the point (1,e) intersects both coordinate axes. What is the area of the triangle formed by this tangent line and the coordinate axes? 2e E) 4e Page 14 of 20
More Differentiation Page 15 25 What is the x coordinate of the point on the curve which is nearest to (4,0)? 1 E) 5 26 A rectangle is to be inscribed in a semicircle of radius 8, with one side lying on On the diameter of the circle. What is the maximum possible area of the rectangle? E) 64 Page 15 of 20
More Differentiation Page 16 27 We need to enclose a field with a fence. We have 500 feet of fencing material and b i i g i i h i w y i g Determine the dimensions of the field that will enclose the largest area. E) 28 If ( ) and the domain is the set of all real x such that then the absolute maximum value of the function f occurs when x is 0 E) 9 Page 16 of 20
More Differentiation Page 17 29 A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the radius of the window be to let in the most light? 3.361 E) 3.784 30 At what approximate rate (in cubic meters per minute) is the volume of a sphere changing at the instant when the surface area is 3 square meters and the radius is increasing at the rate of 1/5 meters per minute? 1.228 E) 0.620 Page 17 of 20
More Differentiation Page 18 31 Point A moves to the right along the positive x-axis at 7 units per second, while point B moves upward along the negative y-axis at 2 units per second. At what rate is the distance between A and B changing when A is at (8,0) and B is at (0,-6)? E) 32 A square is inscribed in a circle. How fast is the area of the square changing when the area of the circle is increasing 1 square inch per minute? (Hint: Diagonals of a square are perpendicular) E) Cannot be determined Page 18 of 20
More Differentiation Page 19 33 A six-foot tall drunk is leaning against a lamp post when his feet start sliding out at a rate of. How fast is his head sliding down the post when his feet are 5 feet out? E) None of the above 34 A funnel is in the shape of a circular cone with the height equal to the diameter, both 6 inches. Liquid is being poured through it at the rate of 2 cubic inches per minute when it becomes clogged. How fast is the level of liquid rising when it is 2 inches deep? E) Page 19 of 20
More Differentiation Page 20 35 If the rate of change of a number x with respect to time t, is x, what is the rate of change of the reciprocal of the number when -16 E) 4 36 A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection? 57.60 E) 67.40 Page 20 of 20