Period: Unit 2 Differentiation Date: mole rat QUIZ 4 Implicit differentiation (calculator OK but unnecessary)

Transcription

1 Unit Differentiation mole rat QUIZ 4 Implicit differentiation (calculator OK but unnecessary) Standard.5: I can find the slope of an equation implicitly (when explicit differentiation is impossible*). *implicit is necessary when it is impossible to cleanly separate the variables (x and y) 1) Find the value of / at (3, 5) ) Write the tangent line at the provided point 3 x y 4x y x y 0x at the point (5,) 3) Write the equation of the tangent line at the point (6,1) 4) Find by implicit differentiation y xy x d y 5) Find (the second derivative) in terms of only x and y: y if 5 x

2 Unit Differentiation (finding the derivatives of functions) moose QUIZ 5 Relate Rates (Calculator OK) any necessary rounding should be to the thousanh (3 decimals) AND only round the final answer. Standard.6a: I can model real world problems with equations and use calculus to analyze the problem. Standard.6b: I can use the concept of implicit differentiation to relate the rate of change of variables with respect to a third variable (the third variable will be invariably time). 1) For the equation Find when y 3x 7x x and when 3 ) A particle is moving along the graph to the right. a) at the point (1,1) the sign of the value of is positive. What is the sign of the value of? b) at the point (,0) the sign of the value of is negative. What is the sign of the value of? c) Which points will the signs of and be the same? i) (,0) ii) (,.449) iii) (0, 3) iv) (,0) v) (1, 3) Circle all the coordinate points where this will be the case

3 3) A (perfectly) spherical balloon is being filled with helium gas at a constant rate of 500 cubic centimeters per minute. (the equation for the volume of a sphere is V r ) a) How long will it take for the balloon to fill with 500 cubic centimeters of helium gas? b) When the radius is 3 centimeters, what is the volume of the balloon? c) When the radius is 3 centimeters, what is the rate of change of the volume of the balloon? d) When the volume is 36 cubic centimeters, what is the rate of change of the radius? 4) A plane travels directly over a radar tower at an altitude of 6 miles. The plane continues flying away from the radar tower at a constant altitude (height). When the plan is 10 miles east of the tower, the radar detects the direct distance between the plane and the tower is changing at a rate of 00 miles per hour. What is the speed of the plane? (*hint*-draw a picture)

4 Unit Differentiation flying squirrel QUIZ 4 Implicit differentiation (calculator OK but unnecessary) Standard.5: I can find the slope of an equation implicitly (when explicit differentiation is impossible*). *implicit is necessary when it is impossible to cleanly separate the variables (x and y) 1) Demonstrate the provided point lies on the curve. ) find the slope of the tangent line at the provided point 3 x y xy 7 at the point (5,) x y 1x at the point (3,) 3) Write the equation of the tangent line at the point (6,1) 4) Find by implicit differentiation y xy x d y 5) Find (the second derivative) in terms of only x and y: y if 3 x

5 Unit Differentiation (finding the derivatives of functions) pterodactyl QUIZ 5 Relate Rates (Calculator OK) any necessary rounding should be to the thousanh (3 decimals) AND only round the final answer. Standard.6a: I can model real world problems with equations and use calculus to analyze the problem. Standard.6b: I can use the concept of implicit differentiation to relate the rate of change of variables with respect to a third variable (the third variable will be invariably time). 1) For the equation Find when y 3x 5x x 3 and when ) A particle is moving along the graph to the right. a) at the point (1,1) the sign of the value of is negative. What is the sign of the value of? b) at the point (,0) the sign of the value of is positive. What is the sign of the value of? c) Which points will the signs of and be opposite of each other? i) (,0) ii) (,.449) iii) (0, 3) iv) (,0) v) (1, 3) Circle all the coordinate points where this will be the case

6 3) A (perfectly) spherical balloon is being filled with helium gas at a constant rate of 00 cubic centimeters per minute. (the equation for the volume of a sphere is V r ) a) How long will it take for the balloon to fill with 1400 cubic centimeters of helium gas? b) When the radius is 3 centimeters, what is the volume of the balloon? c) When the radius is 3 centimeters, what is the rate of change of the volume of the balloon? d) When the volume is 36 cubic centimeters, what is the rate of change of the radius? 4) A plane travels directly over a radar tower at an altitude of 6 miles. The plane continues flying away from the radar tower at a constant altitude (height). When the plan is 10 miles east of the tower, the radar detects the direct distance between the plane and the tower is changing at a rate of 300 miles per hour. What is the speed of the plane? (*hint*-draw a picture)