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Derivatives: Instantaneous Rates of Change & Tangent Line Equations, Assignments of Calculus

Central Washington University (CWU)Calculus

Solutions to various problems related to derivatives, including finding the average rate of change, instantaneous rate of change, and equations of tangent lines for different functions. It covers topics such as finding the derivative using limits, slope of the curve, and the relationship between trigonometric functions and derivatives.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

koofers-user-29s
koofers-user-29s 🇺🇸

10 documents

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bg1
SomeProblemsonDerivatives
1. Let2
3() 2
f
xx=+x
a. Findtheaveragerateofchangeof()
f
xontheinterval[2,5].
b. Findtheinstantaneousrateofchangeof()
f
xat3x
=
.
c. Find'(2)f.
d. If
x
isanynumber,findthevalueof'( )
f
x.
e. Findtheequationofthetangentlinetothecurve()
y
fx
=
at3x
=
.
2. Let1
()fx
x
=
a. Findtheequationofthesecantlinebetweenthepoints1
(1,1) and (3, )
3.
b. Find'(2)f.
c. Findtheequationofthetangentlinetothecurve()
y
fx
=
at2x
.
3. If() sin()
f
xx=,thenforanyvalueof
x
wehave'( ) cos( )
f
xx
=
a. Findtheslopeofthecurvesin( )
y
x
=
atthepoint,1
2
π
⎛⎞
⎜⎟
⎝⎠
.
b. Findtheequationofthetangentlineofthecurvesin( )xy
=
atthepoint,1
2
π
⎛⎞
⎜⎟
⎝⎠
.
4. Let() sin()
f
xx=+x
a. Usethelimitdefinitionofderivativetofindthevalueof'(0)f.
b. Findtheequationofthetangentlinetothecurve()yfx
=
at0x
.
5. Itcanbeshownthat
0
1
lim 1
h
h
e
h
→
−=.
a. Useyourcalculatortocomputethevalueof1
h
e
h
−
for.1,.01,.001,.0001h
=
andseeifthe
valuesappeartobeapproaching1.
b. Usethedefinitionofderivativeasalimittoshowthatif()
x
f
xe
=
,thenforanynumber
awehave'( ) a
f
ae=
h
ee
.(Hint:Recallthatoneofthebasiclawsofexponentssaysthat
+=.)
ah a
e
pf2

Partial preview of the text

Download Derivatives: Instantaneous Rates of Change & Tangent Line Equations and more Assignments Calculus in PDF only on Docsity!

Some Problems on Derivatives

  1. Let 2

f ( ) x = 3 x + 2 x

a. Find the average rate of change of f ( ) x on the interval [2,5].

b. Find the instantaneous rate of change of f ( ) x at x = 3.

c. Find f '(2).

d. If x is any number, find the value of f '( ) x.

e. Find the equation of the tangent line to the curve y = f ( ) x at x = 3.

  1. Let

f ( ) x

x

a. Find the equation of the secant line between the points

(1,1) and (3, )

b. Find f '(2).

c. Find the equation of the tangent line to the curve y = f ( ) x at x = 2.

3. If f ( ) x = sin( ) x , then for any value of x we have f '( ) x =cos( ) x

a. Find the slope of the curve y = sin( ) x at the point ,

⎛ π ⎞

b. Find the equation of the tangent line of the curve y = sin( ) x at the point ,

⎛ π ⎞

4. Let f ( ) x = sin( ) x + x

a. Use the limit definition of derivative to find the value of f '(0).

b. Find the equation of the tangent line to the curve y = f ( ) x at x = 0.

  1. It can be shown that 0

lim 1

h h

e

→ h

a. Use your calculator to compute the value of

h

e

h

for h = .1,.01,.001,.0001and see if the

values appear to be approaching 1.

b. Use the definition of derivative as a limit to show that if ( )

x

f x = e , then for any number

a we have '( )

a

f a = e

h

e e

. (Hint: Recall that one of the basic laws of exponents says that

a h a

e

Answers 0 14 c.

a. 23 b. 2 c.

d. f '( ) x = 6 x + 2.

e. y − 33 = 20( x − 3) or y = 20 x − 27

a. y − 1 = −( 1/ 3)( x −1) or y = −( 1/ 3) x +(4 / 3)

b. − 1/ 4

( 2) or

y x y

− = − = x + 1

a. 0

b. y = 1

a. 2

b. y = 2 x

a. The values of

e − 1

for h .1,.01,.001,.

h

= are approximately given by

h

1.05171,1.00502,1.0005,1.00005. It appears that they are approaching 1. b. By the definition of derivative, we have 0 0 0 0 0

'( ) lim

lim

lim

lim

lim 1

h a h a h a h a h a h h h a a h

f a h f a

f a

h

e e

h

e e e

h

e e

h

e

e e

h

→ + → → → →

= = = ea