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f ( x ) = 2 4 x + 4 x 2

2 4 x + 2 · ln 2 + 8 x

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f ( x ) = x π · π x

π x π 1 · π x + x π · π x ln π

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f ( x ) = ln ( 4 x 3 + x )

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f ( x ) = ln 5 x 7

5 2 ( 5 x 7 )

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f ( x ) = log ( sec x )

tan x ln 10

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f ( x ) = log 7 ( 6 x 4 + 3 ) 5

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f ( x ) = 2 x · log 3 7 x 2 4

2 x · ln 2 · log 3 7 x 2 4 + 2 x · 2 x ln 7 ln 3

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For the following exercises, use logarithmic differentiation to find d y d x .

y = ( sin 2 x ) 4 x

( sin 2 x ) 4 x [ 4 · ln ( sin 2 x ) + 8 x · cot 2 x ]

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y = x log 2 x

x log 2 x · 2 ln x x ln 2

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y = x cot x

x cot x · [ csc 2 x · ln x + cot x x ]

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y = x −1 / 2 ( x 2 + 3 ) 2 / 3 ( 3 x 4 ) 4

x −1 / 2 ( x 2 + 3 ) 2 / 3 ( 3 x 4 ) 4 · [ −1 2 x + 4 x 3 ( x 2 + 3 ) + 12 3 x 4 ]

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[T] Find an equation of the tangent line to the graph of f ( x ) = 4 x e ( x 2 1 ) at the point where

x = −1 . Graph both the function and the tangent line.

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[T] Find the equation of the line that is normal to the graph of f ( x ) = x · 5 x at the point where x = 1 . Graph both the function and the normal line.


The function starts at (−3, 0), decreases slightly and then increases through the origin and increases to (1.25, 10). There is a straight line marked T(x) with slope −1/(5 + 5 ln 5) and y intercept 5 + 1/(5 + 5 ln 5).
y = −1 5 + 5 ln 5 x + ( 5 + 1 5 + 5 ln 5 )

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[T] Find the equation of the tangent line to the graph of x 3 x ln y + y 3 = 2 x + 5 at the point where x = 2 . ( Hint : Use implicit differentiation to find d y d x . ) Graph both the curve and the tangent line.

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Consider the function y = x 1 / x for x > 0 .

  1. Determine the points on the graph where the tangent line is horizontal.
  2. Determine the points on the graph where y > 0 and those where y < 0 .

a. x = e ~ 2.718 b. ( e , ) , ( 0 , e )

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The formula I ( t ) = sin t e t is the formula for a decaying alternating current.

  1. Complete the following table with the appropriate values.
    t sin t e t
    0 (i)
    π 2 (ii)
    π (iii)
    3 π 2 (iv)
    2 π (v)
    2 π (vi)
    3 π (vii)
    7 π 2 (viii)
    4 π (ix)
  2. Using only the values in the table, determine where the tangent line to the graph of I ( t ) is horizontal.
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[T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year.

  1. Write the exponential function that relates the total population as a function of t .
  2. Use a. to determine the rate at which the population is increasing in t years.
  3. Use b. to determine the rate at which the population is increasing in 10 years.

a. P = 500,000 ( 1.05 ) t individuals b. P ( t ) = 24395 · ( 1.05 ) t individuals per year c. 39,737 individuals per year

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[T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present.

  1. Write the exponential function that relates the amount of substance remaining as a function of t , measured in hours.
  2. Use a. to determine the rate at which the substance is decaying in t hours.
  3. Use b. to determine the rate of decay at t = 4 hours.
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[T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1961 is modeled by the function

N ( t ) = 5.3 e 0.093 t 2 0.87 t , ( 0 t 4 ) ,

where N ( t ) gives the number of cases (in thousands) and t is measured in years, with t = 0 corresponding to the beginning of 1960.

  1. Show work that evaluates N ( 0 ) and N ( 4 ) . Briefly describe what these values indicate about the disease in New York City.
  2. Show work that evaluates N ( 0 ) and N ( 3 ) . Briefly describe what these values indicate about the disease in the United States.

a. At the beginning of 1960 there were 5.3 thousand cases of the disease in New York City. At the beginning of 1963 there were approximately 723 cases of the disease in the United States. b. At the beginning of 1960 the number of cases of the disease was decreasing at rate of −4.611 thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of −0.2808 thousand per year.

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[T] The relative rate of change of a differentiable function y = f ( x ) is given by 100 · f ( x ) f ( x ) % . One model for population growth is a Gompertz growth function, given by P ( x ) = a e b · e c x where a , b , and c are constants.

  1. Find the relative rate of change formula for the generic Gompertz function.
  2. Use a. to find the relative rate of change of a population in x = 20 months when a = 204 , b = 0.0198 , and c = 0.15 .
  3. Briefly interpret what the result of b. means.
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For the following exercises, use the population of New York City from 1790 to 1860, given in the following table.

Source: http://en.wikipedia.org/wiki/Largest_cities_in_the_United_States
_by_population_by_decade.
New york city population over time
Years since 1790 Population
0 33,131
10 60,515
20 96,373
30 123,706
40 202,300
50 312,710
60 515,547
70 813,669

[T] Using a computer program or a calculator, fit a growth curve to the data of the form p = a b t .

p = 35741 ( 1.045 ) t

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[T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.

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[T] Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year.

Years since 1790 P
0 69.25
10 107.5
20 167.0
30 259.4
40 402.8
50 625.5
60 971.4
70 1508.5
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[T] Using the tables of first and second derivatives and the best fit, answer the following questions:

  1. Will the model be accurate in predicting the future population of New York City? Why or why not?
  2. Estimate the population in 2010. Was the prediction correct from a.?
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Chapter review exercises

True or False ? Justify the answer with a proof or a counterexample.

Every function has a derivative.

False.

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A continuous function has a continuous derivative.

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A continuous function has a derivative.

False

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If a function is differentiable, it is continuous.

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Use the limit definition of the derivative to exactly evaluate the derivative.

f ( x ) = x + 4

1 2 x + 4

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Find the derivatives of the following functions.

f ( x ) = 3 x 3 4 x 2

9 x 2 + 8 x 3

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f ( x ) = ( 4 x 2 ) 3

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f ( x ) = e sin x

e sin x cos x

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f ( x ) = x 2 cos x + x tan ( x )

x sec 2 ( x ) + 2 x cos ( x ) + tan ( x ) x 2 sin ( x )

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f ( x ) = x 4 sin −1 ( x )

1 4 ( x 1 x 2 + sin −1 ( x ) )

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x 2 y = ( y + 2 ) + x y sin ( x )

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Find the following derivatives of various orders.

First derivative of y = x ln ( x ) cos x

cos x · ( ln x + 1 ) x ln ( x ) sin x

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Third derivative of y = ( 3 x + 2 ) 2

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Second derivative of y = 4 x + x 2 sin ( x )

4 x ( ln 4 ) 2 + 2 sin x + 4 x cos x x 2 sin x

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Find the equation of the tangent line to the following equations at the specified point.

y = cos −1 ( x ) + x at x = 0

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y = x + e x 1 x at x = 1

T = ( 2 + e ) x 2

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Draw the derivative for the following graphs.

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by w ( t ) = 1.9 + 2.9 cos ( π 6 t ) , where t is measured in hours after midnight, and the height is measured in feet.

Find and graph the derivative. What is the physical meaning?

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Find w ( 3 ) . What is the physical meaning of this value?

w ( 3 ) = 2.9 π 6 . At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

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The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

Source: http://news.nationalgeographic.com/news/2005/09/0914_050914_katrina_timeline.html.
Wind speeds of hurricane katrina
Hours after Midnight, August 26 Wind Speed (mph)
1 45
5 75
11 100
29 115
49 145
58 175
73 155
81 125
85 95
107 35

Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?

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Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

−7.5 . The wind speed is decreasing at a rate of 7.5 mph/hr

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Questions & Answers

find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
mujahid
How to do basic integrals
dondi Reply
write something lmit
ram Reply
find the integral of tan tanxdx
Lateef Reply
-ln|cosx| + C
Jug
discuss continuity of x-[x] at [ _1 1]
Atshdr Reply
Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
175000/9*100-100+164294/9*100-100*4
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
66
ram
6
ram
6
Cajab
what is domain in calculus
nelson
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
(-x^5+x^2)/100
Sarada
(-5x^4+2x)/100
Sarada
oh sorry it's (-x^3+1)÷x^2
Misha
-5x^4+2x
Sarada
sorry I didn't understan A with that symbol
Sarada
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
Ga
72x³-72x²+106x-60
okhiria
y= (2x^2+5)(3x+9)^2
lemmor
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
192x^2+50x-1
Daniel
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
Ga
24x²+10x-1
Eyad
eyad Amin that's the correct answer?
Ga
yes
Eyad
ok ok hehe thanks nice dp ekko hahaha
Ga
hahaha 😂❤️❤️❤️ welcome bro ❤️
Eyad
eyad please answer my other question for my assignment
Ga
y= (2x^2+5)(3x+9)^2
lemmor
can i join?
Fernando
yes of course
Jug
can anyone teach me integral calculus?
Jug
it's just the opposite of differential calculus
yhin
of coursr
okhiria
but i think, it's more complicated than calculus 1
Jug
Hello can someone help me with calculus one...
Jainaba
find the derivative of y= (2x+3)raise to 2 sorry I didn't know how to put the raise correctly
Ga Reply
8x+12
Dhruv
8x+3
okhiria
d the derivative of y= e raised to power x
okhiria
Practice Key Terms 1

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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