Solved: State the derivative of each function. (a) (b) (c)

Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

State each differentiation rule both in symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f ) The Quotient Rule (g) The Chain Rule

State the derivative of each function. (a) (b) (c) (d) (e) (f ) (g)

Explain how implicit differentiation works.

(a) Write an expression for the linearization of at . (b) If , write an expression for the differential . (c) If , draw a picture showing the geometric meanings of and .

If is continuous at a, then is differentiable at a.

If and are differentiable, then d dx fx tx f x tx f t f f

If and are differentiable, then d dx  fxtx fxtx

If and are differentiable, then d dx f tx f txtx f t d

If is differentiable, then dx sf x f x 2sfx f 9. I

If is differentiable, then d dx f(sx ) f x 2sx f

d dx tan2 x d dx sec2 x 2,

Calculate y`

Calculate y`

Calculate y`

Calculate y`

Calculate y` y 2xsx 2 1 y

Calculate y` y x 1 x 2  s

Calculate y` y y sincos x t 1 t 2 y 

Calculate y` y sincos x t

Calculate y` y tan s1 x y

Calculate y` y 1 sinx sin x y t

Calculate y` xy 4 x 2 y x 3y y

Calculate y` y sec1 x 2 xy 4 x

Calculate y` y cos y sin 2y xy sec 2 1 tan 2 y

Calculate y` x 2 y cos y sin 2y xy se

Calculate y` y 1 x x sx 1 1 x 2 y

Calculate y` y 1s 3 y 1 x x sx 1 1

Calculate y` sinxy x y ssin sx 2 y y 1

Calculate y` y ssin sx 2

Calculate y` y cot3x 2 5 sin

Calculate y` y x  4 x 4 4 y cot

Calculate y` y sin(tan s1 x 3 ) y

Calculate y` y sin mx x y

Calculate y` y tan x tan y y 1 2 sin  y

Calculate y` x tan y y 1 2

Calculate y` y s 5 x tan x

Calculate y` y x 1x 4 x 2x 3 y s 5 x tan

If , find

40. If , find .

41. Find if .

f f x 12 x n x x 6

Find an equation of the tangent to the curve at the given point. y 4 sin 6, 1 2 x

Find an equation of the tangent to the curve at the given point. y 0, 1 x 2

Find an equation of the tangent to the curve at the given point. y s1 4 sin x 0, 1 y

Find an equation of the tangent to the curve at the given point. x 2, 1 2

At what points on the curve , , is the tangent line horizontal?

Find the points on the ellipse where the tangent line has slope 1.

Suppose that and , where , , , , and . Find (a) and (b) .

If and are the functions whose graphs are shown, let , , and . Find (a) , (b) , and (c) .

Find in terms of . f x x 2 tx f t

Find in terms of . f x tx 2 f x x 2 tx f

Find in terms of . f x  tx 2 f x

Find in terms of . f x x a tx b f x  tx 2 f x tx

Find in terms of . f x t tx f x s

Find in terms of . f x sin tx f x x

Find in terms of . f x tsin x fx t

Find in terms of . fx t(tan sx ) f x

Find in terms of and . hx f xtx fx tx h f t f

Find in terms of and . hx f tsin 4x hx

The graph of is shown. State, with reasons, the numbers at which is not differentiable.

The volume of a right circular cone is , where is the radius of the base and is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant.

A particle moves on a vertical line so that its coordinate at time is , . (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval .

The cost, in dollars, of producing units of a certain commodity is (a) Find the marginal cost function. (b) Find and explain its meaning. (c) Compare with the cost of producing the 101st item.

The volume of a cube is increasing at a rate of 10 . How fast is the surface area increasing when the length of an edge is 30 cm?

A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of , how fast is the water level rising when the water is 5 cm deep?

A balloon is rising at a constant speed of . A boy is cycling along a straight road at a speed of . When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?

A waterskier skis over the ramp shown in the figure at a speed of . How fast is she rising as she leaves the ramp?

The angle of elevation of the Sun is decreasing at a rate of . How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the Sun is ?

(a) Find the linear approximation to near 3. (b) Illustrate part (a) by graphing and the linear approximation. (c) For what values of is the linear approximation accurate to within 0.1?

(a) Find the linearization of at . State the corresponding linear approximation and use it to give an approximate value for . ; (b) Determine the values of for which the linear approximation given in part (a) is accurate to within 0.1

Evaluate if , , and .

A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window.

Express the limit as a derivative and evaluate. lim x l1 x 17 1 x 1 y

Express the limit as a derivative and evaluate. lim h l 0 s 4 16 h 2 h l

Express the limit as a derivative and evaluate. lim  l 3 cos  0.5  3 l

Evaluate s1 tan x s1 sin x x 3

Show that the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant.