100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
grade 12 calculus revision $3.30   Add to cart

Interview

grade 12 calculus revision

1 review
 50 views  1 purchase
  • Course
  • Institution

will help you in exams and tests

Preview 4 out of 61  pages

  • August 25, 2023
  • 61
  • 2023/2024
  • Interview
  • Unknown
  • Unknown
  • 200

1  review

review-writer-avatar

By: astridcassim • 6 months ago

avatar-seller
DIFFERENTIAL CALCULUS

PART 1
(LIMIT, 1ST PRINCIPLE AND
DERIVATIVE USING RULES)


GRADE 12
PAST EXAM PAPERS
EMAILBY
ADDRESS: melulekishabalala@gmail.com
MR M. SHABALALA @NOMBUSO HIGH NUMBER: 0733318802 Page 1
CELLPHONE

, CALCULUS
It helps us to understand any change in different variables ,eg measure instantaneous changes in
rate of population growth, behavior of particles, economics profit and losses also determining the
spread of disease. It is basically the MATHEMATICS OF CHANGE. Wherever there is a
change in one variable produce changes in another, therefore calculus helps us to understand that
changes occur. LIMIT - The limit of the function is the value of 𝑦 to which the graph approaches
as the values of 𝑥 approach a certain value from both the left and right side of that point. It will be
explored more at University.

AVERAGE GRADIENT

The Average gradient (average rate of change) of a function, it is like the gradient of the line
between two points, change in 𝑦 divided by change in 𝑥.

EXAMPLE:

1.1 Find the limit of the following functions.

a) b)


1.2 If 𝑓(𝑥) = −2𝑥 3 find the average gradient between 𝑥 = −1 and 𝑥 = 2

1. “NEW MAIN TOPIC” FIRST PRINCIPLE



 Reads as: Derivative (𝑓′(𝑥)) of 𝑓(𝑥) as ℎ approaches zero.
 General formula to find average gradient(slope) of a function at any point(Derivative)

EXAMPLES OF DERIVATIVE(gradient) USING 1ST PRINCIPLE
1.3 Find 𝑓′(𝑥) from first principle.

1 5
a) 𝑓(𝑥) = 2𝑥 2 − 2 𝑥 b) 𝑓(𝑥) = c) 𝑓(𝑥) = 2𝑥 3 d) 𝑓(𝑥) = 5
𝑥

1.4 In each of the following case find the derivative of 𝑓(𝑥) by 1st principle at the point where 𝑥 = −1
1 2
a) 𝑓(𝑥) = 𝑏 − 𝑥 2 b) 𝑓(𝑥) = 𝑥 2 + 𝑥 − 2 c) 𝑓(𝑥) = − 𝑥 3 d) 𝑓(𝑥) =
2 5𝑥


BY MR M. SHABALALA @NOMBUSO HIGH Page 2

,DIFFERENTIATE THE FOLLOWING BY USING 1ST PRINCIPLE?

a) 𝑓(𝑥) = 𝑥 24 OR b) 𝑓(𝑥) = 𝑥 2 OR c) 𝑓(𝑥) = 3𝑥 2 + 𝑥 12

WOW!  it seems as if it is a challenge now if exponent is too big!
 So far we have dealt with average gradient (gradient between the two or more points)
 Now we will be introduced to the concept of gradient at a point.
 DIFFERENTIATION – Is used to find gradient at a specific point

TWO METHODS OF DETERMINING THE DERIVATIVE
( GRADIENT)
DETERMINE THE DERIVATIVE DETERMINE THE DERIVATIVE USING THE
RULES OF DIFFERENTIATION
FROM FIRST PRINCIPLE
1. Derivative of a constant
2. The power rule
′( )
𝒇(𝒙 + 𝒉) − 𝒇(𝒙) 3. Derivative of a function × a constant
𝒇 𝒙 = lim 4. The sum rule
𝒉→𝟎 𝒉 5. The difference rule


2. THE RULES FOR DIFFERENTIATION
 THE RULES FOR DIFFERENTIATION - allow you to differentiate functions without
going through the process of differentiating from first principles.
 ALWAYS USE RULES TO DIFFERENTIATE UNLESS IF IT STATED TO USE 1ST
PRINCIPLE
RULE 1 RULE 2
 DERIVATIVE OF CONSTANT IS 0  THE POWER RULE
 f(x) = k → ∴ 𝑓 ′ (𝑥) = 0  𝑓(𝑥) = 𝑥 𝑛 →∴ 𝑓 ′ (𝑥) = 𝑛𝑥 𝑛−1
 Eg. f(x)= 𝑥 6 →∴ 𝑓 ′ (𝑥) = 6𝑥 6−1
EXAMPLE f(x) = 5 → ∴ 𝑓 ′ (𝑥) = 0 ∴ 𝑓 ′ (𝑥) = 6𝑥 5
RULE 3
DERIVATIVE OF FUNCTION MULTIPLIED BY A CONSTANT
 𝑓(𝑥) = 𝑘. 𝑔(𝑥) →∴ 𝑓 ′ (𝑥) = 𝑘. 𝑔′ (𝑥)
EXAMPLE F(x) = 5 𝑥 3 →∴ 𝑓 ′ (𝑥) = 5 × 3𝑥 3−1 = 15𝑥 2
RULE 4 RULE 5
 THE SUM RULE  THE DIFFERENCE RULE
 𝑓(𝑥) = 𝑔(𝑥) + ℎ(𝑥)  𝑓(𝑥) = 𝑔(𝑥) − ℎ(𝑥)
→∴ 𝑓 ′ (𝑥) = 𝑔′(𝑥) + ℎ′(𝑥) →∴ 𝑓 ′ (𝑥) = 𝑔(𝑥) − ℎ(𝑥)
EXAMPLE EXAMPLE
𝑓(𝑥) = 4𝑥 2 + 8𝑥 →∴ 𝑓 ′ (𝑥) = 4 × 2𝑥 2−1 + 8 𝑓(𝑥) = 3𝑥 2 − 5𝑥 →∴ 𝑓 ′ (𝑥) = 3 × 2𝑥 2−1 − 5


BY MR M. SHABALALA @NOMBUSO HIGH Page 3

, EXAMPLES OF DIFFERENTIATION USING THE RULES

Before differentiating remove the following:
 Brackets
 Fractions(no variable(e.g x) must be in denominator)
 Surds(√𝑥 remove all radical signs, to exponential form)
 Before differentiate,ensure format ( 𝑦 = 𝑎𝑥 𝑛 𝑜𝑟 𝑦 = 𝑎𝑥 −𝑛 ) NB: 𝑦 = 13 𝑥 2 − 2𝑦
𝑑𝑦
f ′(𝑥) , , 𝐷𝑡 it’s a notation one and same thing. 𝑓 ′′(𝑥) it is a second derivative
𝑑𝑥

1) Given that 𝑓(𝑥) = 2𝑥 3 − 3𝑥 2 + 5 determine 𝑓′(𝑥).
𝑑𝑦
2. Determine if 𝑦 = (2𝑥 + 1)(𝑥 − 3)
𝑑𝑥

𝑡 2 + 5𝑡 + 6
3) Determine 𝐷𝑡 = [ ]
𝑡+2

𝑥2 3
4) Given that 𝑓 (𝑥 ) = + find 𝑓′(𝑥)
3 𝑥3
𝑑 4
5) Determine [ − √𝑥 3 ]
𝑑𝑥 5𝑥 3
𝑑 3
6) Determine [ √8𝑥 2 + (2𝑥 )−4 ]
𝑑𝑥

𝑡 2 + 5𝑡 + 6 (𝒙 + 𝟐)𝟑
7) Determine 𝐷𝑡 = [ ] or Determine 𝑓′(𝑥) if 𝒇(𝒙) =
𝑡 √𝒙

8) Given that 𝑓 (𝜃 ) = 2𝜃 3 − 3𝜃 2 + 5 find 𝑓′′(𝜃)
𝑑
9) Determine [𝑦𝑥 − 𝑦 = 2𝑥 2 − 2𝑥 ]
𝑑𝑥
1
10) Given: 𝑦 = 𝑎𝑥 2 + 𝑎 or determine 𝑓 ′(−1) if 𝑓(𝑥) = 5𝑥 3 + 2𝑟 − 3𝑥3
Determine:
𝑑𝑦
10.1
𝑑𝑥
𝑑𝑦
10.2
𝑑𝑎

BY MR M. SHABALALA @NOMBUSO HIGH Page 4

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller BookSquad. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $3.30. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75057 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
$3.30  1x  sold
  • (1)
  Add to cart