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f ( x ) = 4 x + 7 ; x 1 = 2 , x 2 = 5

4

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f ( x ) = 8 x 3 ; x 1 = −1 , x 2 = 3

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f ( x ) = x 2 + 2 x + 1 ; x 1 = 3 , x 2 = 3.5

8.5

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f ( x ) = x 2 + x + 2 ; x 1 = 0.5 , x 2 = 1.5

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f ( x ) = 4 3 x 1 ; x 1 = 1 , x 2 = 3

3 4

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f ( x ) = x 7 2 x + 1 ; x 1 = −2 , x 2 = 0

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f ( x ) = x ; x 1 = 1 , x 2 = 16

0.2

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f ( x ) = x 9 ; x 1 = 10 , x 2 = 13

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f ( x ) = x 1 / 3 + 1 ; x 1 = 0 , x 2 = 8

0.25

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f ( x ) = 6 x 2 / 3 + 2 x 1 / 3 ; x 1 = 1 , x 2 = 27

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For the following functions,

  1. use [link] to find the slope of the tangent line m tan = f ( a ) , and
  2. find the equation of the tangent line to f at x = a .

f ( x ) = 3 4 x , a = 2

a. −4 b. y = 3 4 x

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f ( x ) = x 5 + 6 , a = −1

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

a. 3 b. y = 3 x 1

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

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f ( x ) = 7 x , a = 3

a. −7 9 b. y = −7 9 x + 14 3

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f ( x ) = 2 3 x 2 , a = −2

a. 12 b. y = 12 x + 14

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f ( x ) = −3 x 1 , a = 4

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f ( x ) = 2 x + 3 , a = −4

a. −2 b. y = −2 x 10

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For the following functions y = f ( x ) , find f ( a ) using [link] .

f ( x ) = 5 x + 4 , a = −1

5

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f ( x ) = −7 x + 1 , a = 3

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f ( x ) = x 2 + 9 x , a = 2

13

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

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f ( x ) = x 2 , a = 6

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f ( x ) = 1 x , a = 2

1 4

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f ( x ) = 1 x 3 , a = −1

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

−3

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For the following exercises, given the function y = f ( x ) ,

  1. find the slope of the secant line P Q for each point Q ( x , f ( x ) ) with x value given in the table.
  2. Use the answers from a. to estimate the value of the slope of the tangent line at P .
  3. Use the answer from b. to find the equation of the tangent line to f at point P .

[T] f ( x ) = x 2 + 3 x + 4 , P ( 1 , 8 ) (Round to 6 decimal places.)

x Slope m P Q x Slope m P Q
1.1 (i) 0.9 (vii)
1.01 (ii) 0.99 (viii)
1.001 (iii) 0.999 (ix)
1.0001 (iv) 0.9999 (x)
1.00001 (v) 0.99999 (xi)
1.000001 (vi) 0.999999 (xii)

a. (i) 5.100000 , (ii) 5.010000 , (iii) 5.001000 , (iv) 5.000100 , (v) 5.000010 , (vi) 5.000001 , (vii) 4.900000 , (viii) 4.990000 , (ix) 4.999000 , (x) 4.999900 , (xi) 4.999990 , (x) 4.999999 b. m tan = 5 c. y = 5 x + 3

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[T] f ( x ) = x + 1 x 2 1 , P ( 0 , −1 )

x Slope m P Q x Slope m P Q
0.1 (i) −0.1 (vii)
0.01 (ii) −0.01 (viii)
0.001 (iii) −0.001 (ix)
0.0001 (iv) −0.0001 (x)
0.00001 (v) −0.00001 (xi)
0.000001 (vi) −0.000001 (xii)
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[T] f ( x ) = 10 e 0.5 x , P ( 0 , 10 ) (Round to 4 decimal places.)

x Slope m P Q
−0.1 (i)
−0.01 (ii)
−0.001 (iii)
−0.0001 (iv)
−0.00001 (v)
−0.000001 (vi)

a. (i) 4.8771 , (ii) 4.9875 (iii) 4.9988 , (iv) 4.9999 , (v) 4.9999 , (vi) 4.9999 b. m tan = 5 c. y = 5 x + 10

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[T] f ( x ) = tan ( x ) , P ( π , 0 )

x Slope m P Q
3.1 (i)
3.14 (ii)
3.141 (iii)
3.1415 (iv)
3.14159 (v)
3.141592 (vi)
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[T] For the following position functions y = s ( t ) , an object is moving along a straight line, where t is in seconds and s is in meters. Find

  1. the simplified expression for the average velocity from t = 2 to t = 2 + h ;
  2. the average velocity between t = 2 and t = 2 + h , where (i) h = 0.1 , (ii) h = 0.01 , (iii) h = 0.001 , and (iv) h = 0.0001 ; and
  3. use the answer from a. to estimate the instantaneous velocity at t = 2 second.

s ( t ) = 1 3 t + 5

a. 1 3 ; b. (i) 0. 3 m/s, (ii) 0. 3 m/s, (iii) 0. 3 m/s, (iv) 0. 3 m/s; c. 0. 3 = 1 3 m/s

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s ( t ) = 2 t 3 + 3

a. 2 ( h 2 + 6 h + 12 ) ; b. (i) 25.22 m/s, (ii) 24.12 m/s, (iii) 24.01 m/s, (iv) 24 m/s; c. 24 m/s

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s ( t ) = 16 t 2 4 t

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Use the following graph to evaluate a. f ( 1 ) and b. f ( 6 ) .

This graph shows two connected line segments: one going from (1, 0) to (4, 6) and the other going from (4, 6) to (8, 8).

a. 1.25 ; b. 0.5

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Use the following graph to evaluate a. f ( −3 ) and b. f ( 1.5 ) .

This graph shows two connected line segments: one going from (−4, 3) to (1, 3) and the other going from (1, 3) to (1.5, 4).
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For the following exercises, use the limit definition of derivative to show that the derivative does not exist at x = a for each of the given functions.

f ( x ) = x 1 / 3 , x = 0

lim x 0 x 1 / 3 0 x 0 = lim x 0 1 x 2 / 3 =

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

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f ( x ) = { 1 , x < 1 x , x 1 , x = 1

lim x 1 1 1 x 1 = 0 1 = lim x 1 + x 1 x 1

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f ( x ) = | x | x , x = 0

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[T] The position in feet of a race car along a straight track after t seconds is modeled by the function s ( t ) = 8 t 2 1 16 t 3 .

  1. Find the average velocity of the vehicle over the following time intervals to four decimal places:
    1. [4, 4.1]
    2. [4, 4.01]
    3. [4, 4.001]
    4. [4, 4.0001]
  2. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at t = 4 seconds.

a. (i) 61.7244 ft/s, (ii) 61.0725 ft/s (iii) 61.0072 ft/s (iv) 61.0007 ft/s b. At 4 seconds the race car is traveling at a rate/velocity of 61 ft/s.

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[T] The distance in feet that a ball rolls down an incline is modeled by the function s ( t ) = 14 t 2 , where t is seconds after the ball begins rolling.

  1. Find the average velocity of the ball over the following time intervals:
    1. [5, 5.1]
    2. [5, 5.01]
    3. [5, 5.001]
    4. [5, 5.0001]
  2. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at t = 5 seconds.
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Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by s = f ( t ) and s = g ( t ) , where s is measured in feet and t is measured in seconds.

Two functions s = g(t) and s = f(t) are graphed. The first function s = g(t) starts at (0, 0) and arcs upward through roughly (2, 1) to (4, 4). The second function s = f(t) is a straight line passing through (0, 0) and (4, 4).
  1. Which vehicle has traveled farther at t = 2 seconds?
  2. What is the approximate velocity of each vehicle at t = 3 seconds?
  3. Which vehicle is traveling faster at t = 4 seconds?
  4. What is true about the positions of the vehicles at t = 4 seconds?

a. The vehicle represented by f ( t ) , because it has traveled 2 feet, whereas g ( t ) has traveled 1 foot. b. The velocity of f ( t ) is constant at 1 ft/s, while the velocity of g ( t ) is approximately 2 ft/s. c. The vehicle represented by g ( t ) , with a velocity of approximately 4 ft/s. d. Both have traveled 4 feet in 4 seconds.

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[T] The total cost C ( x ) , in hundreds of dollars, to produce x jars of mayonnaise is given by C ( x ) = 0.000003 x 3 + 4 x + 300 .

  1. Calculate the average cost per jar over the following intervals:
    1. [100, 100.1]
    2. [100, 100.01]
    3. [100, 100.001]
    4. [100, 100.0001]
  2. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.
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[T] For the function f ( x ) = x 3 2 x 2 11 x + 12 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the ZOOM feature on the calculator to approximate the two values of x = a for which m tan = f ( a ) = 0 .

a.
The function starts in the third quadrant, passes through the x axis at x = −3, increases to a maximum around y = 20, decreases and passes through the x axis at x = 1, continues decreasing to a minimum around y = −13, and then increases through the x axis at x = 4, after which it continues increasing.
b. a 1.361 , 2.694

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[T] For the function f ( x ) = x 1 + x 2 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the ZOOM feature on the calculator to approximate the values of x = a for which m tan = f ( a ) = 0 .
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Suppose that N ( x ) computes the number of gallons of gas used by a vehicle traveling x miles. Suppose the vehicle gets 30 mpg.

  1. Find a mathematical expression for N ( x ) .
  2. What is N ( 100 )? Explain the physical meaning.
  3. What is N ( 100 ) ? Explain the physical meaning.

a. N ( x ) = x 30 b. 3.3 gallons. When the vehicle travels 100 miles, it has used 3.3 gallons of gas. c. 1 30 . The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled 100 miles.

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[T] For the function f ( x ) = x 4 5 x 2 + 4 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate f ( −2 ) , f ( −0.5 ) , f ( 1.7 ) , and f ( 2.718 ) .
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[T] For the function f ( x ) = x 2 x 2 + 1 , do the following.

  1. Use a graphing calculator to graph f in an appropriate viewing window.
  2. Use the nDeriv function on a graphing calculator to find f ( −4 ) , f ( −2 ) , f ( 2 ) , and f ( 4 ) .

a.
The function starts in the second quadrant and gently decreases, touches the origin, and then it increases gently.
b. −0.028 , −0.16 , 0.16 , 0.028

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Questions & Answers

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
explain for me
Usman
okay I have such documents
Fitzgerald
please share it
Hamza
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how en where can u apply it
Migos
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier Reply
Practice Key Terms 4

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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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