Deck 2: Derivatives

Question 1

Use differentials to estimate the amount of paint needed to apply a coat of paint cm thick to a hemispherical dome with diameter m.

A)
B)
C)
D)
E)

Answer 1

Question 2

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s how fast is the boat approaching the dock when it is m from the dock? Round the result to the nearest hundredth if necessary.

Answer 2

Question 3

Find dy/dx by implicit differentiation.

Answer 3

Question 4

If two resistors with resistances and are connected in parallel, as in the figure, then the total resistance measured in ohms ( $\Omega$ ), is given by .If and are increasing at rates of and respectively, how fast is changing when and ? Round the result to the nearest thousandth.

A)
B)
C)
D)
E)

Question 5

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0.

A)
B)
C)
D)
E)

Question 6

Find the differential of the function.

A)
B)
C)
D)
E)

Question 7

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth.

Question 8

The circumference of a sphere was measured to be cm with a possible error of cm. Use differentials to estimate the maximum error in the calculated volume.

Question 9

Gravel is being dumped from a conveyor belt at a rate of ft/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is ft high? Round the result to the nearest hundredth.

A)
B)
C)
D)
E)

Question 10

A plane flying horizontally at an altitude of 1 mi and a speed of mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

A)
B)
C)
D)
E)

Question 11

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure below). The point Q is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

Question 12

Calculate .

A)
B)
C)
D)
E)none of these

Question 13

The top of a ladder slides down a vertical wall at a rate of m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

A)
B)
C)
D)
E)None of these

Question 14

Use the linear approximation of the function at to approximate the number .

A)
B)
C)
D)
E)

Question 15

If a snowball melts so that its surface area decreases at a rate of , find the rate at which the diameter decreases when the diameter is cm.

Question 16

The curve with the equation is called an asteroid. Find an equation of the tangent to the curve at the point ( , 1).

Question 17

Find the tangent line to the ellipse at the point .

A)
B)
C)
D)
E)

Question 18

Find an equation of the tangent line to the given curve at the indicated point.

Question 19

Two cars start moving from the same point. One travels south at mi/h and the other travels west at mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

A)
B)
C)
D)
E)

Question 20

Two sides of a triangle are m and m in length and the angle between them is increasing at a rate of rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\Omega$ .

A)
B)
C)
D)
E)

Question 21

If find the domain of

A)
B)
C)
D)
E)

Question 22

Differentiate.

A)
B)
C)
D)
E)

Question 23

Find an equation of the tangent line to the curve at the point $\Omega$
.

Question 24

Find the derivative of the function. = 3 sin x + 9x - 9

A)-3 sin x + 9
B)3 cos x + 9
C)3 sin x + 9
D)-3 cos x + 9

Question 25

Find equations of the tangent lines to the curve that are parallel to the line .

A)
B)
C)
D)
E)

Question 26

If , find .

A)
B)
C)
D)
E)

Question 27

Differentiate.

Question 28

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 29

Find the derivative of the function.

Question 30

Differentiate.

Question 31

If , find and .

A)
B)
C)
D)
E)

Question 32

Find the limit.

Question 33

Differentiate.

Question 34

Find an equation of the tangent line to the curve.

Question 35

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

A)
B)
C)
D)
E)

Question 36

If find the defination of derivative to find

A)
B)
C)
D)
E)None of these

Question 37

Find the given derivative by finding the first few derivatives and observing the pattern that occurs.

A)
B)
C)
D)
E)None of these

Question 38

Suppose that and , , , and .Find .

A)140
B)
C)20
D)17
E)24

Question 39

Differentiate. $\Omega$
$\Omega$

Question 40

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 41

Use the definition of the derivative to find

A)
B)
C)
D)
E)does not exist

Question 42

Find the slope of the tangent line to the curve at the point .

A)
B)
C)
D)
E)

Question 43

If a cylindrical tank holds gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume of water remaining in the tank after t minutes as Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t.

Question 44

If an equation of the tangent line to the curve at the point where

A)
B)
C)
D)
E)None of these

Question 45

If , find .

A)
B)
C)
D)
E)

Question 46

Find an equation of the tangent line to the parabola at the point .

A)
B)
C)
D)
E)

Question 47

The cost (in dollars) of producing x units of a certain commodity is Find the instantaneous rate of change with respect to x when (This is called the marginal cost.)

A)
B)
C)
D)
E)

Question 48

The cost (in dollars) of producing x units of a certain commodity is Find the average rate of change with respect to x when the production level is changed from

Question 49

Find the derivative of the function.

Question 50

Find an equation of the tangent line to the curve at the point .

A)
B)
C)
D)
E)None of these