With Anand.Duration:
Notes
CPT Exam Exposure
With Anand.Duration:
Notes
CPT Exam Exposure
CPT Chapter Differential Calculus
With Anand.Duration:4hrs 42 mins
| Differentiation | Duration (min:sec) |
| {modal https://www.youtube.com/embed/UvyoqUciuas?autoplay=1;rel=0|width=780|height=439|title=What is Calculus}What is Calculus{/modal} | 15:32 |
| Differentiation | |
| {modal https://www.youtube.com/embed/MlNpdRUL-10?autoplay=1;rel=0|width=780|height=439|title=Index}Index{/modal} | 09:42 |
| {modal https://www.youtube.com/embed/25E25J6AYaY?autoplay=1;rel=0|width=780|height=439|title=What is Differentiation}What is Differentiation{/modal} | 09:11 |
| {modal https://www.youtube.com/embed/rrVI14yEe4M?autoplay=1;rel=0|width=780|height=439|title=Note- They all are different}Note- They all are different{/modal} | 04:00 |
| {modal https://www.youtube.com/embed/6EnMJClpKzw?autoplay=1;rel=0|width=780|height=439|title=Simple Formula}Simple Formula{/modal} | 05:05 |
| {modal https://www.youtube.com/embed/qvciLAzTLl8?autoplay=1;rel=0|width=780|height=439|title=Differentiation of a Constant}Differentiation of a Constant{/modal} | 01:56 |
| {modal https://www.youtube.com/embed/OIvriQLGRDU?autoplay=1;rel=0|width=780|height=439|title=Understanding Simple Formula}Understanding Simple Formula{/modal} | 07:11 |
| {modal https://www.youtube.com/embed/l3yVdR0QNLM?autoplay=1;rel=0|width=780|height=439|title=Problems on Simple Formula}Problems on Simple Formula{/modal} | 07:15 |
| {modal https://www.youtube.com/embed/aID7CiwTyBk?autoplay=1;rel=0|width=780|height=439|title=Addition & Subtraction Rule}Addition & Subtraction Rule{/modal} | 09:31 |
| {modal https://www.youtube.com/embed/wZgRGBV2wT8?autoplay=1;rel=0|width=780|height=439|title=Problems on Addition Rule}Problems on Addition Rule{/modal} | 15:49 |
| {modal https://www.youtube.com/embed/2giYLKUHvuI?autoplay=1;rel=0|width=780|height=439|title=Quotient Rule}Quotient Rule{/modal} | 13:48 |
| {modal https://www.youtube.com/embed/DkqIpesoFTY?autoplay=1;rel=0|width=780|height=439|title=Problem 1 QR}Problem 1 QR{/modal} | 15:18 |
| {modal https://www.youtube.com/embed/TyvZFwLnb5g?autoplay=1;rel=0|width=780|height=439|title=Problem 2 QR}Problem 2 QR{/modal} | 25:41 |
| {modal https://www.youtube.com/embed/_J94pZuERI4?autoplay=1;rel=0|width=780|height=439|title=Chain Rule}Chain Rule{/modal} | 09:19 |
| {modal https://www.youtube.com/embed/fMII21N_c7U?autoplay=1;rel=0|width=780|height=439|title=Problem 1 CR}Problem 1 CR{/modal} | 26:06 |
| {modal https://www.youtube.com/embed/s7vl0bkCgv0?autoplay=1;rel=0|width=780|height=439|title=Problem 2 CR}Problem 2 CR{/modal} | 14:36 |
| {modal https://www.youtube.com/embed/WvwC-18GLYk?autoplay=1;rel=0|width=780|height=439|title=Log Function}Log Function{/modal} | 22:41 |
| {modal https://www.youtube.com/embed/6tzFuzdyWig?autoplay=1;rel=0|width=780|height=439|title=Log Problem}Log Problem{/modal} | 12:55 |
| {modal https://www.youtube.com/embed/ZTHs2lRQWZE?autoplay=1;rel=0|width=780|height=439|title=Parametric Function}Parametric Function{/modal} | 04:19 |
| {modal https://www.youtube.com/embed/lf2n5mLgq7E?autoplay=1;rel=0|width=780|height=439|title=Problem Parametric Function}Problem Parametric Function{/modal} | 09:18 |
| {modal https://www.youtube.com/embed/M_ycfNtaxwY?autoplay=1;rel=0|width=780|height=439|title=Implicit Function}Implicit Function{/modal} | 04:55 |
| {modal https://www.youtube.com/embed/iQWzfTbMfkw?autoplay=1;rel=0|width=780|height=439|title=Problem Implicit Function}Problem Implicit Function{/modal} | 08:05 |
| {modal https://www.youtube.com/embed/_3uSFjCUgYk?autoplay=1;rel=0|width=780|height=439|title=Second Derivative}Second Derivative{/modal} | 01:46 |
| {modal https://www.youtube.com/embed/2j9VRTMcU4w?autoplay=1;rel=0|width=780|height=439|title=Problem Second Derivative}Problem Second Derivative{/modal} | 08:51 |
| {modal https://www.youtube.com/embed/DCsA9rdVH_k?autoplay=1;rel=0|width=780|height=439|title=Misc. Problem}Misc. Problem{/modal} | 14:27 |
| {modal https://www.youtube.com/embed/7kLEvAHgCAc?autoplay=1;rel=0|width=780|height=439|title=Conclusion}Conclusion{/modal} | 05:24 |
| Total | 04:42:41 |
What is Calculus?
Arithmetic is a study of quantity
Geometry study shapes
Trigonometry study traingle
Calculus studies change
Branches of Calculus-
Differentiation or Derivatives
Integration
Differentiation-
What is Differentiation?
Simple Formula
Addition & Subtraction Rule
Product & Quotient Rule
Chain Rule
Logarithmic Expression
Parametric Function
Implicit Function
Second derivatives
Simple Formulaxn
ex
ax
xx
Examples-
If f(x)=xk and f '(1)=10, then the value of k is :
(a) 10
(b) -10
(c) 1/10
(d) None
If xy = 1 then y2 + dy/dx is equal to
a) 1
b) 0
c) –1
d) none of these
Addition & Subtraction-
The slope of the tangent to the curve y = x2 –x at the point, where the line y = 2 cuts the curve in the First quadrant, is
a) 2 b) 3 c) –3 d) none of these
For the curve x2 + y2 + 2gx + 2hy = 0, the value of dy/dx at (0, 0) is
a) -g/h
b) g/h
c) h/g
d) none of these
Product & Quotient Rule-
If x = y log (xy), then dy/dx equal to:
(a) x +y/x (1+logxy)
(b) x - y/x (1+logxy)
(c) x +y/x (logx+logy)
(d) x - y/x (logx+logy)
If xy (x-y)=0, find dy/dx :
(a) y(2x - y)/x(2y - x)
(b) x(2x - y)/y(2y - x)
(c) y(2y - x) /x(2x - y)
(d) None of these
The slope of the tangent at the point (2,-2) to the curve
x2+ xy+y2-4 = 0 is given by :
(a) 0
(b) 1
(c) - 1
(d) None
Chain Rule-
If x3y2=(x-y)5.Find dy/dx at (1,2).
(a)-7/9
(b) 7/9
(c) 9/7
(d) - 9/7
If y=log(5-4x2/3+5x2) , then dy = dx
(a) 8/(4x-5)-10/(3+5x)
(b) (4x2-5)-(3+5x2)
(c) -8x/(4x2-5)-10x/(3+5x2)
(d) 8x-10
If (x2/a2) - (y2/a2) = 1, (dy/dx) can be expressed as
a) x/a
b)x/√(x2-a2)
c){1/(x2/a2)-1}
d) none of these
If log (x / y) = x + y, (dy/dx) may be found to be
a)y(1-x)/x(1+y)
b)y/x
c)(1-x)/(1+y)
d)none of these
Log Functions-
Rules of Log-
Parametric Functions
y= 15t , x=log(t3+t)
y= 5t4+3 , x=4t
x=2t+5and y=t2-5,then dx/dy=?
(a)t
(b)-1/t
(c)1/t
(d)0
If x=ct, y=c/t, then dy/dx is equal to:
(a) 1/t
(b) t.et
(c) -1/t2
(d) None of these
Implicit Functions-
xy+y2x+15=0
If xy=yx, then dy/dx gives :
(a) x(x logy-y)/ y(ylogx-x)
(b) x(y Iogx-x)/ y(xlogy-y)
(c) y(x logy- y)/ x(ylogx-x)
(d) None of these
Second Derivative-
Find the second derivative of y=√x+1
(a) 1/2 (x +1)-1/2
(b) -1/4 (x+1)-3/2
(c) 1/4 (x +1)-1/2
(d) None of these
If y=2x+4/x, then x2 d2y/dx2+xdy/dx - y yields
(a) 3
(b) 1
(c) 0
(d) 4
Conclusion-
What is Differentiation?
Simple Formula
Addition & Subtraction Rule
Product & Quotient Rule
Chain Rule
Logarithmic Expression
Parametric Function
Implicit Function
Second derivatives
Notes
CPT Exam Exposure
CPT Chapter Integral Calculus
With Anand.Duration:5 hrs 26mins
| Intergration | Duration (min:sec) |
| {modal https://www.youtube.com/embed/-EeUyEvk4EI?autoplay=1;rel=0|width=780|height=439|title=Introduction}Introduction{/modal} | 08:36 |
| {modal https://www.youtube.com/embed/mP2_XQjW_vc?autoplay=1;rel=0|width=780|height=439|title=Index}Index{/modal} | 06:24 |
| {modal https://www.youtube.com/embed/d5N4njlrGeQ?autoplay=1;rel=0|width=780|height=439|title=What is Integration}What is Integration{/modal} | 03:23 |
| {modal https://www.youtube.com/embed/inXZV4JUB7Q?autoplay=1;rel=0|width=780|height=439|title=Basic Formulae}Basic Formulae{/modal} | 11:14 |
| {modal https://www.youtube.com/embed/qjNg4FkSG7Q?autoplay=1;rel=0|width=780|height=439|title=Integration of Constant}Integration of Constant{/modal} | 03:40 |
| {modal https://www.youtube.com/embed/H8mtr8c2RB0?autoplay=1;rel=0|width=780|height=439|title=Example Basic Formulae}Example Basic Formulae{/modal} | 06:06 |
| {modal https://www.youtube.com/embed/ajHI_PH1lsE?autoplay=1;rel=0|width=780|height=439|title=Problems on Basic Formulae}Problems on Basic Formulae{/modal} | 15:18 |
| {modal https://www.youtube.com/embed/NV7vYDekeDY?autoplay=1;rel=0|width=780|height=439|title=Special Variable}Special Variable{/modal} | 25:56 |
| {modal https://www.youtube.com/embed/yVqLaEHPMyw?autoplay=1;rel=0|width=780|height=439|title=Method of Substitution & Examples}Method of Substitution & Examples{/modal} | 15:07 |
| {modal https://www.youtube.com/embed/pDHcexG6Svw?autoplay=1;rel=0|width=780|height=439|title=Problems on Method of Substitution}Problems on Method of Substitution{/modal} | 26:29 |
| {modal https://www.youtube.com/embed/IsMgMWgJsEY?autoplay=1;rel=0|width=780|height=439|title=Integration by Parts & Example}Integration by Parts & Example{/modal} | 11:18 |
| {modal https://www.youtube.com/embed/wxqZ-iyk17M?autoplay=1;rel=0|width=780|height=439|title=Problems Integration by Parts}Problems Integration by Parts{/modal} | 19:29 |
| {modal https://www.youtube.com/embed/GL44NPGvxF0?autoplay=1;rel=0|width=780|height=439|title=Special cases- Integration by Parts}Special cases- Integration by Parts{/modal} | 06:17 |
| {modal https://www.youtube.com/embed/LZl-4e55YsA?autoplay=1;rel=0|width=780|height=439|title=Partial Fraction}Partial Fraction{/modal} | 08:52 |
| {modal https://www.youtube.com/embed/ciDOn5JQYEY?autoplay=1;rel=0|width=780|height=439|title=Problem on Partial Fraction}Problem on Partial Fraction{/modal} | 21:00 |
| {modal https://www.youtube.com/embed/wFuW_H7GrUw?autoplay=1;rel=0|width=780|height=439|title=Special Substitution}Special Substitution{/modal} | 28:07 |
| {modal https://www.youtube.com/embed/EjJYakuzKvY?autoplay=1;rel=0|width=780|height=439|title=What is Definite Integration (DI)}What is Definite Integration (DI){/modal} | 03:54 |
| {modal https://www.youtube.com/embed/h6YyPm905Vk?autoplay=1;rel=0|width=780|height=439|title=Properties of DI}Properties of DI{/modal} | 06:23 |
| {modal https://www.youtube.com/embed/hAll5LsoK2Q?autoplay=1;rel=0|width=780|height=439|title=Examples of DI}Examples of DI{/modal} | 06:14 |
| {modal https://www.youtube.com/embed/cInjzCoaW9Y?autoplay=1;rel=0|width=780|height=439|title=Problems 1 DI}Problems 1 DI{/modal} | 26:07 |
| {modal https://www.youtube.com/embed/86vqD3nSoYA?autoplay=1;rel=0|width=780|height=439|title=Problems 2 DI}Problems 2 DI{/modal} | 21:06 |
| {modal https://www.youtube.com/embed/VMYYBQLkABI?autoplay=1;rel=0|width=780|height=439|title=Problems 3 DI}Problems 3 DI{/modal} | 20:35 |
| {modal https://www.youtube.com/embed/K6Dw_1cCDgA?autoplay=1;rel=0|width=780|height=439|title=Slope Problem}Slope Problem{/modal} | 06:06 |
| {modal https://www.youtube.com/embed/obS9tSRxRkE?autoplay=1;rel=0|width=780|height=439|title=Mis Problem}Mis Problem{/modal} | 12:57 |
| {modal https://www.youtube.com/embed/fsFeWan8azI?autoplay=1;rel=0|width=780|height=439|title=Summary}Summary{/modal} | 05:28 |
| Total | 5:26:06 |
Few things which we have covered
What is Integration?
Basic Formulae
Special Variable
Actual Substitution
Integration by parts
Parts Special case
Partial Fraction
Special Substitution
Definite Integration
Properties of Definite Integration
What is Integration?
Integration is reverse of Differentiation
ex-
Symbol
∫x2dx
Integration of a constant-
∫ a dx =∫a.x0 dx
∫ 7 dx =
Addition & Subtraction to a constant-
3+k
3-k
Example-
∫(√x +1/√x)dx
(a) 2x1/2(1/3 x-1)
(b) 2x1/2(1/3 x+1)
(c) 2(1/3x + x1/2)
(d) None of these.
Evaluate = ∫5x2 dx :
(a) 5/3x3+ k (b) 5x3/3+k (c) 5x3 (d) none of these
Integration of 3 – 2x – x4 will become
(a) – x2 – x5 / 5 (b) 3x - x2 - x5 /5+ k (c) 3x -x2 +x5/5 +k (d) none of these
Given f(x) = 4x3 + 3x2 – 2x + 5 and ∫ f(x) dx is
(a) x4 + x3 – x2 + 5x (b) x4 + x3 – x2 + 5x + k
(c) 12x2 + 6x – 2x2 (d) none of these
Special Variables-
This is applied when by taking a linear function of x i.e ax or ax+b as X , we can apply any of the standard formulae
Here we apply standard formula and divide the answer with derivative of ax+b i.e a
Use method of substitution to integrate the function f(x)=(4x+5)6 and the answer is
(a) 1/28(4x+5)7+k
(b) (4x+5)7/7+k
(c) (4x+5)7/7
(d) none of these
Integrate (x+a)n and the result will be
(a) (x+a)n+1/n+1 + k
(b) (x+a)n+1/n+1
(c) (x+a)n+1
(d) none of these
Method of Substitution or Change of Variable-
Substitution
When we have a function & it’s derivatives
∫ (5x2+4x+3)4 (10x+4)dx
Note - Trick to identify which one is function & which is derivative
Evaluate: ∫dx/√(x2+a2)
(a) 1/2 - log (x+√x2+a2) + C
(b) log (x+√x2+a2) +C
(C) log (x √x2+a2)+ C
(d) 1/2log (x √x2+a2)+ C
Square Partial Integration-
Using method of partial fraction to evaluate
∫ (x+5) dx/(x +1) (x + 2)2 we get
(a) 4 log (x + 1) - 4log(x + 2)+3/x+2+k
(b) 4 log (x + 2) – 3/x +2)+k
(c) 4 log (x + 1) – 4 log(x+2)
(d) none of these
Special Substitution-
∫d( x2+1) / √x2+2 is equal to
(a) x/2 (√x2 +2)+k
(b) √x2 +2 +k
(c) 1/(x2+2)3/2 +k
(d) none of these
Definite Integration-
The value of ∫(1+logx)/x dx is:[Given Loge =1]
(a) 1/2
(b) 3/2
(c) 1
(d) 5/2
Summary-
What is Integration?
Basic Formulae
Special Variable
Actual Substitution
Integration by parts
Parts Special case
Partial Fraction
Special Substitution
Definite Integration
Properties of Definite Integration
Notes
CPT Exam Exposure
CPT Chapter Limits and Continuity
With Anand.Duration:4hrs25min
| Limits and Continuity | Duration (min:sec) |
| Limits | |
| {modal https://www.youtube.com/embed/fMa4J8ExCeE?autoplay=1;rel=0|width=780|height=439|title=Overview}Overview{/modal} | 07:49 |
| {modal https://www.youtube.com/embed/lkwOYbOV-ng?autoplay=1;rel=0|width=780|height=439|title=What is Limit}What is Limit{/modal} | 18:21 |
| {modal https://www.youtube.com/embed/9VtwhxEUaeY?autoplay=1;rel=0|width=780|height=439|title=Limit with graph}Limit with graph{/modal} | 03:35 |
| {modal https://www.youtube.com/embed/AqHhhtsDy18?autoplay=1;rel=0|width=780|height=439|title=Basics of Limit}Basics of Limit{/modal} | 24:27 |
| {modal https://www.youtube.com/embed/RSzHZWr7Ks4?autoplay=1;rel=0|width=780|height=439|title=Avoid Indeterminant1}Avoid Indeterminant1{/modal} | 24:41 |
| {modal https://www.youtube.com/embed/ucsvlv-cSfI?autoplay=1;rel=0|width=780|height=439|title=Avoid Indeterminant2}Avoid Indeterminant2{/modal} | 22:02 |
| {modal https://www.youtube.com/embed/3GYVhL9ENUs?autoplay=1;rel=0|width=780|height=439|title=How to solve infinity problem}How to solve infinity problem{/modal} | 02:14 |
| {modal https://www.youtube.com/embed/QWPzznaOQdU?autoplay=1;rel=0|width=780|height=439|title=Infinity Problem}Infinity Problem{/modal} | 21:03 |
| {modal https://www.youtube.com/embed/fhIVjBntGCg?autoplay=1;rel=0|width=780|height=439|title=Limit Formulae}Limit Formulae{/modal} | 06:19 |
| {modal https://www.youtube.com/embed/WeooWVNneOE?autoplay=1;rel=0|width=780|height=439|title=Formulae Example1}Formulae Example1{/modal} | 16:42 |
| {modal https://www.youtube.com/embed/LqXwrhEnFgw?autoplay=1;rel=0|width=780|height=439|title=Formulae Example2}Formulae Example2{/modal} | 29:09 |
| {modal https://www.youtube.com/embed/KBlsOrmVNiQ?autoplay=1;rel=0|width=780|height=439|title=Odd & Even Function}Odd & Even Function{/modal} | 07:07 |
| Continuity | |
| {modal https://www.youtube.com/embed/yvCwDeMELnI?autoplay=1;rel=0|width=780|height=439|title=Continuity Concept}Continuity Concept{/modal} | 14:25 |
| {modal https://www.youtube.com/embed/n28k1RpIoLw?autoplay=1;rel=0|width=780|height=439|title=Function Discontinuous}Function Discontinuous{/modal} | 03:54 |
| {modal https://www.youtube.com/embed/l9TwJI8MAME?autoplay=1;rel=0|width=780|height=439|title=Checking Limit Exist Concept}Checking Limit Exist Concept{/modal} | 03:57 |
| {modal https://www.youtube.com/embed/9XafNpJUn1I?autoplay=1;rel=0|width=780|height=439|title=Example Checking Limit Exist}Example Checking Limit Exist{/modal} | 12:58 |
| {modal https://www.youtube.com/embed/JuE0PNRN6dU?autoplay=1;rel=0|width=780|height=439|title=Check Continuity-Concept}Check Continuity-Concept{/modal} | 02:40 |
| {modal https://www.youtube.com/embed/21rggi6rwew?autoplay=1;rel=0|width=780|height=439|title=Check Countinity-Example1}Check Countinity-Example1{/modal} | 17:28 |
| {modal https://www.youtube.com/embed/OsXigh0yNNg?autoplay=1;rel=0|width=780|height=439|title=Check Countinity-Example2}Check Countinity-Example2{/modal} | 23:18 |
| {modal https://www.youtube.com/embed/ye-RDos1kF4?autoplay=1;rel=0|width=780|height=439|title=Conclusion}Conclusion{/modal} | 02:37 |
| Total | 04:24:46 |
What are Limits?
Basic of Limit
Avoid Indeterminant
Infinity Problem
Formula based Problem
Condition for discontinuous
Check Limit Exist or Not
Check Continuous or Not
Avoid Indeterminant-
lim (4x4+5x3+7x2+6x)/(5x5+7x2+x) is equal to
x →0
a) 6 b) 5 c) -6 d) none of these
lim(x2-5x+6)(x2-3x+2)/(x3-3x2+4) is equal to
x →2
a)1/3 b)3 c) -⅓ ) none of these
lim (4-x2)/3-√(x2+5) is equal to
x →2
a) 6 b) 1/6 c) –6 d) none of these
lim (x2-1)/(√3x+1-√5x-1) is evaluated to be
x →1
a) 4 b) ¼ c) -4 d)none of these
Formulae-
f(x) ex-1 = loge = logee =1
x →0 x
f(x) ax-1 = loga = logea
x →0 x
f(x) log(1+x) = loge = 1
x →0 x
f(x) loga(1+x) = logae
x →0 x
f(x) log(1+1/x) = loge = 1
x →∞ 1/x
f(x) (1+1/x )x=e
x →∞
f(x) (1+x )1/x=e
x →∞
Formula based Problem-
lim (e2x-1)/x is equal to
x →0
a)1/2
(b) 2
(c) e5
(d) none of these.
Lim x→0 (2e1/x -3x)/(e1/x +x)=-
a)-3
b)0
c)2
d)4
lim (5x +3x -2)/x will be equal to
x →0
a) loge15 b) log (1/15) c) log e d) none of these
Odd & Even Function-
if f(x) is an odd function then
a){f(-x)+f(x)}/2 is an even function
b){|x|+1} is even when [x] = the integer x≤
c) [f(x)+f(-x)]/2 is neither even or odd
d) none of these
Continuity-
When a function is discontinuous-
A function will be discontinuous when the function is indeterminate.
The points of discontinuity of the function :
f(x) = (2x2+6x-5)/(12x2+x-20) are, when x is
a)-4/5 and 5/3
b)-4/3 and 5/4
c)4/5 and -5/3
d)4/3 and -5/4
Checking Limit Exist or Not-
If f(x)=(x+1)/√(6x2+3)+3x then lim f(x) and f(-1)
x →-1
a) both exists
b) one exists and other does not exist
c) both do not exists
d)none of these
lim y→0 (3y+ |y|)/(7y -5|y|)=
a)2
b)⅙
c)3/7
d)does not exist
Checking Continuity-
A function f(x) is defined as follows
f(x) = x2 when 0 < x <1
= x when 1 <_ x < 2
= (1/4) x3 when 2 < x < 3
Now f(x) is continuous at
a) x = 1 b) x = 3 c) x = 0 d) none of these.
Conclusion-
What are Limits?
Basic of Limit
Avoid Indeterminant
Infinity Problem
Formula based Problem
Condition for discontinuous
Check Limit Exist or Not
Check Continuous or Not
Notes
CPT Exam Exposure