2.2 - Problem #8 (Calc)

Cory Sheeley
24 Sept 202003:08
EducationalLearning
32 Likes 10 Comments

TLDRThis video script discusses problem number eight from section 2.2, which involves evaluating the limit of a function as h approaches zero, a common calculus concept. The function given is f(x) = x^2, and the script guides through the process of finding the slope of the tangent line by simplifying the expression f(x + h) - f(x)/h. The cancellation of terms and factorization lead to the conclusion that the slope at x = -3 is -6, demonstrating the derivative's application in determining the tangent's slope at a specific point.

Takeaways
  • ๐Ÿ“š The problem discusses a common calculus concept: the limit of the form \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\), which is used to find the slope of the tangent line to a curve at a given point.
  • ๐Ÿ” The specific function given in the problem is \(f(x) = x^2\), and the goal is to evaluate the limit as \(h\) approaches 0 for a given \(x\) value.
  • ๐Ÿ“˜ The process begins by substituting \(x + h\) into the function, resulting in \((x + h)^2\).
  • ๐Ÿงฉ The expansion of \((x + h)^2\) is done, yielding \(x^2 + 2xh + h^2\).
  • โœ‚๏ธ The next step involves subtracting \(x^2\) from the expanded form, simplifying the expression to \(2xh + h^2\).
  • ๐Ÿ”„ The script then factors out an \(h\) from the terms \(2xh + h^2\), preparing for the limit operation.
  • ๐Ÿ“‰ As \(h\) approaches 0, the terms involving \(h\) in the numerator cancel out, leaving \(2x\) as the result of the limit.
  • ๐Ÿ“Œ The value of \(x\) is specified as \(-3\), and substituting this into the simplified limit gives \(2 \times (-3)\), which is \(-6\).
  • ๐Ÿ“ The final takeaway is that the slope of the tangent line to the function \(f(x) = x^2\) at \(x = -3\) is \(-6\).
  • ๐Ÿ“ The process is a clear demonstration of how to find the derivative of a function at a specific point, which is a fundamental concept in calculus.
Q & A
  • What is the main concept being discussed in the script?

    -The script discusses the concept of finding the slope of a tangent line to a curve at a given point using the limit of a function as h approaches zero.

  • What is the specific limit form being evaluated in the script?

    -The limit form being evaluated is \( \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \), which represents the derivative of the function f at point x.

  • What function is being used to demonstrate the concept in the script?

    -The function used in the script is \( f(x) = x^2 \).

  • What is the value of x for which the limit is being evaluated?

    -The limit is being evaluated for \( x = -3 \).

  • How does the script simplify the expression for the limit?

    -The script simplifies the expression by substituting \( f(x) \) with \( x^2 \), expanding \( (x + h)^2 \), and then canceling out the \( x^2 \) terms.

  • What is the result of the simplified expression before taking the limit?

    -The simplified expression before taking the limit is \( \frac{2xh + h^2}{h} \).

  • What happens when the limit is taken as h approaches zero?

    -As h approaches zero, the terms \( 2xh \) and \( h^2 \) in the numerator become negligible compared to \( h \) in the denominator, leaving \( 2x \) as the result of the limit.

  • What is the final value of the limit after plugging in x = -3?

    -After plugging in \( x = -3 \), the final value of the limit is \( 2 \times (-3) = -6 \).

  • What does the final value of the limit represent in the context of the function?

    -The final value of the limit represents the slope of the tangent line to the function \( f(x) = x^2 \) at the point where \( x = -3 \).

  • Why is the limit form important in calculus?

    -The limit form is important in calculus because it is the foundation for understanding derivatives, which describe the rate of change of a function and the slopes of tangent lines at any point on the function's graph.

  • Can the process described in the script be applied to any differentiable function?

    -Yes, the process described in the script can be applied to any differentiable function to find the slope of its tangent line at a specific point.

Outlines
00:00
๐Ÿ“š Calculus Limit Problem Explanation

This paragraph introduces a common calculus problem involving the limit of a function as h approaches zero. The main objective is to find the slope of the tangent line to the function at a given point x. The problem is presented in the context of the limit of the form \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). The function provided is f(x) = x^2, and the process involves substituting x with (x+h), expanding the squared term, and simplifying the expression to isolate the limit. The cancellation of terms and factoring out of h leads to the discovery that the limit is 2x, which represents the slope of the tangent line.

Mindmap
Derivative
Rate of Change
Tangent Line
Final Answer
Specific Calculation
Slope of the Tangent
Limit Evaluation
Simplification
Expansion
Substitution
Given Value of x
Given Function
Importance in Calculus
Definition of Limit
Conceptual Understanding
Result Interpretation
Calculation Process
Problem Statement
Introduction to Limits
Calculus: Limits and Derivatives
Alert
Keywords
๐Ÿ’กLimit
In calculus, a 'limit' refers to the value that a function or sequence approaches as the input approaches some value. In the context of the video, the limit is used to find the slope of the tangent line to a curve at a specific point, which is a fundamental concept in understanding rates of change.
๐Ÿ’กFunction
A 'function' is a mathematical relationship between two sets that assigns to each element from the first set exactly one element of the second set. In the video, the function 'f(x)' is used to represent a mathematical rule that assigns a unique output for each input value, specifically 'x squared' in this case.
๐Ÿ’กSlope
The 'slope' of a line represents its steepness or incline. In the video, the slope is being calculated to find the rate of change of the function at a particular point, which is done by evaluating the limit of the difference quotient as 'h' approaches zero.
๐Ÿ’กTangent Slope
The 'tangent slope' is the slope of the tangent line to a curve at a given point. It is calculated using the limit process described in the video, which is a way to find the instantaneous rate of change of a function at a specific point.
๐Ÿ’กDifference Quotient
The 'difference quotient' is an expression used to estimate the slope of the secant line between two points on a curve. In the video, the difference quotient is simplified to find the limit, which gives the slope of the tangent line at the point x.
๐Ÿ’กDistribute
To 'distribute' in algebra means to multiply each term inside a set of parentheses by an expression outside the parentheses. In the script, the term '(x + h) squared' is distributed to find its expanded form, which is a step in simplifying the expression to find the limit.
๐Ÿ’กFactor
To 'factor' is to express a polynomial as the product of its factors. In the video, the term '2xh + h^2' is factored by taking 'h' common, which simplifies the expression and helps in finding the limit.
๐Ÿ’กApproach Zero
When a variable 'approaches zero', it means it gets arbitrarily close to zero but never actually equals zero. In the context of the video, 'h approaches zero' is used to find the limit of the expression, which is a standard technique in calculus to determine the behavior of functions at specific points.
๐Ÿ’กCancel
To 'cancel' in the context of algebraic expressions means to eliminate terms that are equal but have opposite signs. In the script, the 'x squared' terms cancel each other out, simplifying the expression and making it easier to find the limit.
๐Ÿ’กEvaluate
To 'evaluate' a mathematical expression means to calculate its value. In the video, the expression is evaluated by substituting 'x' with a specific value (-3) to find the slope of the function at that point.
๐Ÿ’กInstantaneous Rate of Change
The 'instantaneous rate of change' is the rate at which a quantity changes at a specific instant in time. In the video, the limit process is used to find this rate for the function at the point x = -3, which is a key concept in differential calculus.
Highlights

Problem 8 for section 2.2 involves evaluating a limit that finds the tangent slope.

The limit is of the form \( \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \), which is common in calculus.

The function given is \( f(x) = x^2 \).

The process starts by substituting \( x + h \) into the function, resulting in \( (x + h)^2 \).

Expanding \( (x + h)^2 \) gives \( x^2 + 2xh + h^2 \).

The next step is to subtract \( x^2 \) from the expanded form.

The expression simplifies to \( \frac{2xh + h^2}{h} \) after cancellation.

Factoring out an \( h \) from the numerator simplifies the expression further.

The limit as \( h \) approaches 0 is then considered.

The \( h \) terms cancel out, leaving \( 2x \).

The value of \( x \) is given as -3, which is substituted into the expression.

The slope of the tangent line is found to be 2 at \( x = -3 \).

The final calculation results in a slope of -6 for the function at \( x = -3 \).

The process demonstrates the calculation of the derivative as a limit.

The problem illustrates the concept of the derivative as the slope of the tangent line.

The solution provides a clear example of applying calculus to find the slope at a specific point.

The transcript explains the steps in a logical and methodical manner.

Transcripts
00:00

this is problem number eight for section

00:02

2.2

00:03

and this problem says limits of the form

00:06

limit

00:06

as h approaches zero of f of

00:09

x plus h minus f of x over h occur

00:12

frequently in calculus

00:14

and we've already seen these in section

00:15

two one

00:17

evaluate this limit for the given value

00:19

of x

00:20

and function f so what they want you to

00:23

do is they want you to find essentially

00:25

the slope because that's really what

00:26

this does is it finds the tangent slope

00:30

so let's start by taking and using the

00:32

equation so limit

00:34

as h approaches 0 of

00:37

f of x plus h minus

00:40

f of x all over h

00:43

and we'll start by plugging in

00:48

our function and we end up with x

00:52

i'm plugging x plus h into x squared so

00:55

it's x plus h

00:57

squared minus the function itself well

01:00

that's just

01:01

x squared all over h

01:05

i'm going to go ahead and distribute x

01:07

plus h squared

01:08

so it's x plus h times x plus h and i

01:11

get

01:12

x times x which is x squared x

01:15

times h h times x

01:18

so i get plus x h

01:21

and plus x h

01:25

plus h times h which is h squared so

01:28

this

01:29

is x squared plus 2 x h

01:32

plus h squared

01:37

now i'm going to take this and i'm going

01:39

to plug it in for

01:40

x plus h squared so you end up with

01:43

equals limit as

01:46

h approaches 0

01:50

x squared plus 2xh

01:54

plus h squared minus x squared

01:58

all over h the x squared

02:02

and the negative x squared cancel

02:05

and then i can actually factor an h out

02:07

of these two terms

02:09

and i'm going to get limit as h

02:12

approaches 0

02:14

of when i factor an h out i get h

02:17

that'll leave you with 2x plus

02:21

h on top and that's over h

02:24

which gives you the limit of h

02:28

approaches zero or as

02:29

h approaches zero this will cancel and

02:32

i'm left with two x

02:33

plus h

02:37

i'm going to plug in 0 and when i get

02:39

that i get

02:40

2x now they want us to evaluate this or

02:44

find the slope so the slope

02:47

is going to equal 2 and this is at

02:51

x equal negative 3 so i'll write that

02:53

out so this is going to equal 2

02:55

times negative 3 which is negative

02:58

6 for our solution that's the slope

03:02

for this x squared function at x equal

03:05

negative 3.