<< Chapter < Page Chapter >> Page >

Find the derivative of h ( x ) = cos −1 ( 3 x 1 ) .

h ( x ) = −3 6 x 9 x 2

Got questions? Get instant answers now!

Applying the inverse tangent function

The position of a particle at time t is given by s ( t ) = tan −1 ( 1 t ) for t 1 2 . Find the velocity of the particle at time t = 1 .

Begin by differentiating s ( t ) in order to find v ( t ) . Thus,

v ( t ) = s ( t ) = 1 1 + ( 1 t ) 2 · −1 t 2 .

Simplifying, we have

v ( t ) = 1 t 2 + 1 .

Thus, v ( 1 ) = 1 2 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the equation of the line tangent to the graph of f ( x ) = sin −1 x at x = 0 .

y = x

Got questions? Get instant answers now!

Key concepts

  • The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
  • We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

Key equations

  • Inverse function theorem
    ( f −1 ) ( x ) = 1 f ( f −1 ( x ) ) whenever f ( f −1 ( x ) ) 0 and f ( x ) is differentiable.
  • Power rule with rational exponents
    d d x ( x m / n ) = m n x ( m / n ) 1 .
  • Derivative of inverse sine function
    d d x sin −1 x = 1 1 ( x ) 2
  • Derivative of inverse cosine function
    d d x cos −1 x = −1 1 ( x ) 2
  • Derivative of inverse tangent function
    d d x tan −1 x = 1 1 + ( x ) 2
  • Derivative of inverse cotangent function
    d d x cot −1 x = −1 1 + ( x ) 2
  • Derivative of inverse secant function
    d d x sec −1 x = 1 | x | ( x ) 2 1
  • Derivative of inverse cosecant function
    d d x csc −1 x = −1 | x | ( x ) 2 1

For the following exercises, use the graph of y = f ( x ) to

  1. sketch the graph of y = f −1 ( x ) , and
  2. use part a. to estimate ( f −1 ) ( 1 ) .
A curved line starting at (−2, 0) and passing through (−1, 1) and (2, 2).

a.
A curved line starting at (−3, 0) and passing through (−2, 1) and (1, 2). There is another curved line that is symmetric with this about the line x = y. That is, it starts at (0, −3) and passes through (1, −2) and (2, 1).
b. ( f −1 ) ( 1 ) ~ 2

Got questions? Get instant answers now!
A quarter circle starting at (0, 4) and ending at (4, 0).

a.
A quarter circle starting at (0, 4) and ending at (4, 0).
b. ( f −1 ) ( 1 ) ~ 1 / 3

Got questions? Get instant answers now!

For the following exercises, use the functions y = f ( x ) to find

  1. d f d x at x = a and
  2. x = f −1 ( y ) .
  3. Then use part b. to find d f −1 d y at y = f ( a ) .

f ( x ) = 6 x 1 , x = −2

Got questions? Get instant answers now!

f ( x ) = 2 x 3 3 , x = 1

a. 6, b. x = f −1 ( y ) = ( y + 3 2 ) 1 / 3 , c. 1 6

Got questions? Get instant answers now!

f ( x ) = 9 x 2 , 0 x 3 , x = 2

Got questions? Get instant answers now!

f ( x ) = sin x , x = 0

a. 1 , b. x = f −1 ( y ) = sin −1 y , c. 1

Got questions? Get instant answers now!

For each of the following functions, find ( f −1 ) ( a ) .

f ( x ) = x 2 + 3 x + 2 , x −1 , a = 2

Got questions? Get instant answers now!

f ( x ) = x 3 + 2 x + 3 , a = 0

1 5

Got questions? Get instant answers now!

f ( x ) = x 2 x , x < 0 , a = 1

1 3

Got questions? Get instant answers now!

f ( x ) = x + sin x , a = 0

Got questions? Get instant answers now!

f ( x ) = tan x + 3 x 2 , a = 0

1

Got questions? Get instant answers now!

For each of the given functions y = f ( x ) ,

  1. find the slope of the tangent line to its inverse function f −1 at the indicated point P , and
  2. find the equation of the tangent line to the graph of f −1 at the indicated point.

f ( x ) = 4 1 + x 2 , P ( 2 , 1 )

Got questions? Get instant answers now!

f ( x ) = x 4 , P ( 2 , 8 )

a. 4 , b. y = 4 x

Got questions? Get instant answers now!

f ( x ) = ( x 3 + 1 ) 4 , P ( 16 , 1 )

Got questions? Get instant answers now!

f ( x ) = x 3 x + 2 , P ( −8 , 2 )

a. 1 96 , b. y = 1 13 x + 18 13

Got questions? Get instant answers now!

f ( x ) = x 5 + 3 x 3 4 x 8 , P ( −8 , 1 )

Got questions? Get instant answers now!

For the following exercises, find d y d x for the given function.

y = sin −1 ( x 2 )

2 x 1 x 4

Got questions? Get instant answers now!

y = sec −1 ( 1 x )

−1 1 x 2

Got questions? Get instant answers now!

y = ( 1 + tan −1 x ) 3

3 ( 1 + tan −1 x ) 2 1 + x 2

Got questions? Get instant answers now!

y = cos −1 ( 2 x ) · sin −1 ( 2 x )

Got questions? Get instant answers now!

y = 1 tan −1 ( x )

−1 ( 1 + x 2 ) ( tan −1 x ) 2

Got questions? Get instant answers now!

y = cot −1 4 x 2

x ( 5 x 2 ) 4 x 2

Got questions? Get instant answers now!

For the following exercises, use the given values to find ( f −1 ) ( a ) .

f ( π ) = 0 , f ( π ) = −1 , a = 0

−1

Got questions? Get instant answers now!

f ( 6 ) = 2 , f ( 6 ) = 1 3 , a = 2

Got questions? Get instant answers now!

f ( 1 3 ) = −8 , f ( 1 3 ) = 2 , a = −8

1 2

Got questions? Get instant answers now!

f ( 3 ) = 1 2 , f ( 3 ) = 2 3 , a = 1 2

Got questions? Get instant answers now!

f ( 1 ) = −3 , f ( 1 ) = 10 , a = −3

1 10

Got questions? Get instant answers now!

f ( 1 ) = 0 , f ( 1 ) = −2 , a = 0

Got questions? Get instant answers now!

[T] The position of a moving hockey puck after t seconds is s ( t ) = tan −1 t where s is in meters.

  1. Find the velocity of the hockey puck at any time t .
  2. Find the acceleration of the puck at any time t .
  3. Evaluate a. and b. for t = 2 , 4 , and 6 seconds.
  4. What conclusion can be drawn from the results in c.?

a. v ( t ) = 1 1 + t 2 b. a ( t ) = −2 t ( 1 + t 2 ) 2 c. ( a ) 0.2 , 0.06 , 0.03 ; ( b ) 0.16 , −0.028 , −0.0088 d. The hockey puck is decelerating/slowing down at 2, 4, and 6 seconds.

Got questions? Get instant answers now!

[T] A building that is 225 feet tall casts a shadow of various lengths x as the day goes by. An angle of elevation θ is formed by lines from the top and bottom of the building to the tip of the shadow, as seen in the following figure. Find the rate of change of the angle of elevation d θ d x when x = 272 feet.

A building is shown with height 225 ft. A triangle is made with the building height as the opposite side from the angle θ. The adjacent side has length x.
Got questions? Get instant answers now!

[T] A pole stands 75 feet tall. An angle θ is formed when wires of various lengths of x feet are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle d θ d x when a wire of length 90 feet is attached.

A flagpole is shown with height 75 ft. A triangle is made with the flagpole height as the opposite side from the angle θ. The hypotenuse has length x.

−0.0168 radians per foot

Got questions? Get instant answers now!

[T] A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by θ = tan −1 ( x 2000 ) , where x is the height of the rocket. Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.

A rocket is shown with in the air with the distance from its nose to the ground being x. A triangle is made with the rocket height as the opposite side from the angle θ. The adjacent side has length 2000.
Got questions? Get instant answers now!

[T] A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle θ (in radians) given by θ = cot −1 x 40 cot −1 x 10 ,

where x is the distance in feet away from the movie screen that the person is sitting, as shown in the following figure.

A person is shown with a right triangle coming from their eye (the right angle being on the opposite side from the eye), with height 10 and base x. There is a line drawn from the eye to the top of the screen, which makes an angle θ with the triangle’s hypotenuse. The screen has a height of 30.
  1. Find d θ d x .
  2. Evaluate d θ d x for x = 5 , 10 , 15 , and 20.
  3. Interpret the results in b..
  4. Evaluate d θ d x for x = 25 , 30 , 35 , and 40
  5. Interpret the results in d. At what distance x should the person stand to maximize his or her viewing angle?

a. d θ d x = 10 100 + x 2 40 1600 + x 2 b. 18 325 , 9 340 , 42 4745 , 0 c. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening. d. 54 12905 , 3 500 , 198 29945 , 9 1360 e. As the person moves beyond 20 feet from the screen, the viewing angle is decreasing. The optimal distance the person should stand for maximizing the viewing angle is 20 feet.

Got questions? Get instant answers now!

Questions & Answers

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
ayo Reply
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
Ahmad
by using integration product formula
Roha
find derivative f(x)=1/x
Mul Reply
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
ZIMBA Reply
do u mean fundamental theorem ?
Roha
I want simple integral
aparna Reply
for MSc chemistry... simple formulas of integration
aparna
hello?
funny
how are you
funny
I don't understand integration
aparna
r u insane
aparna
integration is so simple not typical..
funny
tell me any questions about integration then i will solve.
funny
we use integration for whole values or for sum of values any there are some basic rule for integration..
funny
I just formulas
aparna
I just want formulas of integration
aparna
value of log ax cot-x cos-x
aparna
there are many formulas about integration
funny
more then one formula are exist about integration..
funny
so I want simple formulas Because I'm studying MSc chem...Nd have done bsc from bio...
aparna
I am M.sc physics now i am studying in m.phil
funny
so what can i do
aparna
I will send you basic formula for integration after two mint first of all i write then i will send you.
funny
send me your messenger id where i can send you formulas about integration because there is no option for image sending..
funny
integration f(X) dx this is basic formula of integration sign is not there you can look integration sign in methematics form... and f(X) my be any function any values
funny
you send me your any ID where i can send you information about integration
funny
send me SMS at this ID Adnan sathi Adnan sathi
funny
Hi
RIZWAN
I don't understand the formula
Adaeze Reply
who's formula
funny
which formula?
Roha
what is the advantages of mathematical economics
Mubarak
calculus 1 ke. solutions kahan se millyn gy ? koi bta de 😭😭
Hamza

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask