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MAT265 Review Problems for Exam 2
Product and Quotient Rules ( )( 1. Suppose ( ) 2. Suppose the derivative of ( )
( )
) with ( ) exists. Assume that ( )
( ) ( )
Find ( ) Let ( )
( )
( ) at a. Find an equation of the tangent line to ( ) at b. Find an equation of the tangent line to 3. Suppose tangent to at is and tangent to line tangent to the following curves at ( ) ( ) a. b.
( ) ( )
at
is
Find the
Chain Rule using a table )) and ( ) ( 4. Let ( ) ( ( )) , ( ) ( ( )) , ( ) ( ( ( ) ) . Using the table to compute the following derivatives. a. ( ) b. ( ) c. ( ) d. ( ) e. ( ) f. ( ) 0 -1 1 2 -2 1 0 5 4 2 2 3 3 5 10 3 5 -5 1 20 4 1 -8 3 15 5 0 -10 2 20
( ) ( ) ( ) ( )
Derivative of composite functions 5. Suppose is differentiable on [-2, 2] with ( ) Evaluate the followings. a. ( ) b. ( ) ( ) c. ( ) ( ) ( ) . Let ( )
Other Chain Rule 6. Solve. a. Suppose [ ( )] b. If Horizontal tangent 7. Find all points on [ ] at which ( ) has horizontal tangent. and ( ) and ( ) , find when
Find
( ).
Tangent line using Implicit Differentiation 8. Find an equation of the tangent line to the curve at the given point. a. ( ) b. c. ( Second Derivatives 9. Find a. b. Related Rates 10. Sand falls from an overhead bin at a rate of 3 and accumulates in a cone shaped pile, whose height is equal to twice the radius. How fast is the height of the cone rising when the height is 2 meters? 11. Two small planes approach an airport, one flying due west at 120 mi/hr and the other flying due north at 150 mi/hr. Assuming they fly at the same constant elevation, how fast is the distance between the planes changing when the westbound plane is 180 mi from the airport and the northbound plane is 225 mi from the airport. 12. If a snowball melts so that its surface area decreases at the rate of , find the rate at which the diameter decreases when the diameter is 10 cm. (Hint : Surface area ) 13. A 13-ft ladder is leaning against a vertical wall when Jack begins
Product and Quotient Rules ( )( 1. Suppose ( ) 2. Suppose the derivative of ( )
( )
) with ( ) exists. Assume that ( )
( ) ( )
Find ( ) Let ( )
( )
( ) at a. Find an equation of the tangent line to ( ) at b. Find an equation of the tangent line to 3. Suppose tangent to at is and tangent to line tangent to the following curves at ( ) ( ) a. b.
( ) ( )
at
is
Find the
Chain Rule using a table )) and ( ) ( 4. Let ( ) ( ( )) , ( ) ( ( )) , ( ) ( ( ( ) ) . Using the table to compute the following derivatives. a. ( ) b. ( ) c. ( ) d. ( ) e. ( ) f. ( ) 0 -1 1 2 -2 1 0 5 4 2 2 3 3 5 10 3 5 -5 1 20 4 1 -8 3 15 5 0 -10 2 20
( ) ( ) ( ) ( )
Derivative of composite functions 5. Suppose is differentiable on [-2, 2] with ( ) Evaluate the followings. a. ( ) b. ( ) ( ) c. ( ) ( ) ( ) . Let ( )
Other Chain Rule 6. Solve. a. Suppose [ ( )] b. If Horizontal tangent 7. Find all points on [ ] at which ( ) has horizontal tangent. and ( ) and ( ) , find when
Find
( ).
Tangent line using Implicit Differentiation 8. Find an equation of the tangent line to the curve at the given point. a. ( ) b. c. ( Second Derivatives 9. Find a. b. Related Rates 10. Sand falls from an overhead bin at a rate of 3 and accumulates in a cone shaped pile, whose height is equal to twice the radius. How fast is the height of the cone rising when the height is 2 meters? 11. Two small planes approach an airport, one flying due west at 120 mi/hr and the other flying due north at 150 mi/hr. Assuming they fly at the same constant elevation, how fast is the distance between the planes changing when the westbound plane is 180 mi from the airport and the northbound plane is 225 mi from the airport. 12. If a snowball melts so that its surface area decreases at the rate of , find the rate at which the diameter decreases when the diameter is 10 cm. (Hint : Surface area ) 13. A 13-ft ladder is leaning against a vertical wall when Jack begins