Calculus Maximus WS 4.1: Tangent Line Problem

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Worksheet 4.1—Tangent Line Problem

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1. Find the slope of the tangent lines to the graphs of the following functions at the indicated points. Usethe alternate form.(a) ( ) 3 2f x x= − at ( )1,5− (b) ( ) 25g x x= − at 2x =

2. Find the derivative function, either ( )f x! or dydx

, of each of the following using the limit definition.

(a) ( ) 22 3 4f x x x= + − (b) 31

yx

=−

(c) ( ) 2f x x= −

Calculus Maximus WS 4.1: Tangent Line Problem

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3. For each of the following, find the equation of BOTH the tangent line and the normal line to the

function at the indicated points. Use either limit definition of the derivative. (a) ( ) 2 1g x x= + at ( )2,5 (b) 1y x= − at 9x = 4. Find an equation of the line that is tangent to ( ) 3f x x= and parallel to the line 3 1 0x y− + =

Calculus Maximus WS 4.1: Tangent Line Problem

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5. For each of the following functions ( )f x , on the same coordinate grid, sketch a graph of ( )f x! . (a) (b) (c)

(d) (e) (f)

(g)

6. Sketch a function that has the following characteristics. ( )0 4f = , ( )0 0f ! = , ( ) 0f x! < for 0x < , and ( ) 0f x! > for 0x >

Calculus Maximus WS 4.1: Tangent Line Problem

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7. If ( ) 3f c! = , find ( )f c! − if

(a) f is an odd function, and if (b) f is an even function. 8. Find the equations of the two tangent lines to ( ) 2f x x= that pass through the point ( )1, 3− , a point not

on the graph of f. Hint: Sketch the scenario first. 9. For each of the following, the limit represents ( )f c! for a function ( )f x and a number x c= . Find

both f and c.

(a) ( )

0

5 3 1 2limh

hh→

− + −# $% & (b) ( )3

0

2 8limh

hh→

− + +

(c) 2

6

36lim6x

xx→

− +

− (d)

9

2 6lim9x

xx→

−

−

Calculus Maximus WS 4.1: Tangent Line Problem

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10. Use alternate form to discuss the differentiability of each of the following at the point.

(a) ( ) 3 2f x x x= + at 1x = (b) ( )( )

( )

3

2

1 , 1

1 , 1

x xf x

x x

! − ≤$= %

− >$&

at 1x =

(c) ( )2 1, 2

4 3, 2x xf xx x

! + ≤= #

− >% at 2x = (d) ( )

1 1, 22

2 , 2

x xf x

x x

! + <"= #" ≥%

at 2x =

Calculus Maximus WS 4.1: Tangent Line Problem

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11. True or False. If false, explain why or give a counterexample.

(a) The slope of the tangent line to the differentiable function f at the point ( )( )2, 2f is

( ) ( )2 2f h fh

+ −.

(b) If a function is continuous at a point, then it is differentiable at that point.

(c) If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

(d) If a function is differentiable at a point, then it is continuous at that point.