12th CBSE 2020 - MATHEMATICS : 9 Days 80/80 Plan · 2020. 3. 10. · 12th CBSE 2020 - MATHEMATICS :...
Transcript of 12th CBSE 2020 - MATHEMATICS : 9 Days 80/80 Plan · 2020. 3. 10. · 12th CBSE 2020 - MATHEMATICS :...
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12th CBSE 2020 - MATHEMATICS : 9 Days 80/80 Plan
Name of Units Weightage
Inverse Trigonometry ( Ch-2 ) 7th Mar (COMPLETED) 08
Calculus ( Ch- 5,6,7,8,9 ) Ch-5 ( COMPLETED) 8, 9, 10th Mar 35
Vectors and Three Dimensional Geometry ( Ch -10, 11) 11th Mar 14
Linear Programming ( Ch - 12) 12th Mar 05
Probability ( Ch- 13) 10th Mar 08
Relations and Functions ( Ch-1 ) 12th Mar 08
Algebra - Matrices and Determinants ( Chs 3 & 4) 13th Mar 10
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2015 2016 2017 2018 2019 2020(Sample)
Very Short Answer (1 mark) 1 Q
Short Answer (2 marks) 2 Q 1 Q 1 Q
Long Answer Type I (4 marks)
1 Q 1 Q 1 Q
Long Answer Type II (6 marks)
1 Q 1 Q 1 Q 1 Q 1 Q 1 Q
Application of Derivatives TREND for LAST 5 Years
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The ratio ΔyΔx is average rate of change of ‘y’ with respect to ‘x’
Derivative as a Rate Measure
where Δx represents change in ‘x’& Δy represents change in ‘y’
Now, if we apply the limit Δx → 0 represents
inst. rate of change of ‘y’
w.r.t. x, at x = x0
dydx x0
lim ΔyΔx = dy
dxΔx→0
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Class 12th - 4 Marks (CBSE 2017)
Q1. The volume of a cube is increasing at the rate of 9 cm3/s.How fast is its surface area increasing when the length of an edge is 10 cm?
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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Q2. The total cost C(x) associated with the product of x units of an item is given by Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of cost at any level of output.
Class 12th - 4 Marks (CBSE 2018)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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TANGENTS AND NORMALS
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Hence, equation of a tangent at (x0 , y0) is
TangentsWe know that = f ′(x)
dydx
f ′(x0) = Slope of tangent at (x0, y0)
y – y0 = f ′(x0)(x – x0)
P(x0, y0)
y = f(x)
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Hence, the equation of a Normal at (x0 , y0) is
Normals
Normal is perpendicular to Tangent
–1 /f ′(x0)=Slope of Normal at (x0, y0)
y – y0 = –1 /f ′(x0) (x – x0)
P(x0, y0)
y = f(x)
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Q3. Find the equations of the tangent and the normal, to the curve 16x2+9y2=145 at the point (x1, y1), where x1=2 and y1>0 Class 12th - 4 Marks (CBSE 2016)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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SOLUTION
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Q4. Find the equation of tangents to the curve y=x3+2x - 4, which are perpendicular to line x+14y+3=0
Class 12th - 4 Marks (CBSE 2016)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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Q5. Find the equation of tangent to curves x = sin 3t, y = cos2t at .
(CBSE 2016 - 4 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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Q6. Find the equation of tangent to the curve at the
point, where it cuts the X-axis.
(CBSE 2010 - 4 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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Q7. Find the equations of tangents to the curve 3x2 - y2 = 8, which passes
through the point .
(CBSE 2013 - 4 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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SOLUTION
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A function f (x) is said to be increasing if
A function f (x) is said to be decreasing if
INCREASING AND DECREASING FUNCTIONS
x2
> x1 ⇒ f (x
2)
> f (x
1)
A function f (x) is said to be non decreasing if
x2
> x1 ⇒
f (x
2) ≥
f (x
1)
x2
> x1 ⇒
f (x
2) <
f (x
1)
A function f (x) is said to be non increasing if
x2
> x1 ⇒
f(x
2) ≤
f(x
1)
x2
x1
f(x1)
f(x2)
f(x1)
f(x2)
x2
x1
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DERIVATIVE TEST
If function is differentiable then
f ′(x) > 0 ⇒ f is increasing f ′(x) < 0 ⇒ f is decreasing
Let us study a test to check if the function is increasing or decreasing
We need to take care that set of values for which f ′(x) = 0 do not form an interval.
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Q8. Find the intervals in which the function
is (a) Strictly increasing, (b) Strictly decreasing.
(CBSE 2018 - 4 marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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Q9. Find the values(s) of x for which y= [x(x-2)]2 is an increasing function.
(CBSE 2014 - 4 marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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Q10. Prove that is an increasing function of θ on
(CBSE 2016 - 4 marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
MAXIMA AND MINIMA
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12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
MAXIMA AND MINIMA
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Q11. Show that the surface area of a closed cuboid with square base and given volume is minimum, when is a cube.
(CBSE 2017 - 4 marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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Q12. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is . Also show that the maximum volume of the cone is of the volume of the sphere.
CBSE 2014
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Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is . Also show that the maximum volume of the cone is of the volume of the sphere.
CBSE 2014Solution
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Solution
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Solution
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Solution
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Q13. If the length of three sides of a trapezium other than the base are each equal to 10 cm, then find the area of the trapezium, when it is maximum.
(CBSE 2014, 2010 - 6 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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SOLUTION
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Q14. A manufacturer can sell x items at a price of Rs each.The cost of x items is Rs .Find the number of items he should sell to reach maximum profit.
(CBSE 2009 - 6 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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Errors and Approximations
Approximation is, knowing the value of a function at one point and using it to decide the value of the function at another point.
Error: If error in x is Δx, and y is a function of x, what will be the error in y ?
E.g. We know the value of f(2) And we approximate the value of f(2.001)
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However, if Δx is small, then
We know, if Δx → 0, =ΔyΔx
dydx
then the quantity
ΔyΔx
≈dydx
Δy ≈dydx
×Δx
f(a + h) = f(a) + h.f ′(a) It is used to find f(a + h);knowing f(a) and f ′(a)
Smaller the magnitude of Δx , better is theapproximation
f ′ (a) =f(a + h) – f(a)
h1)
2)
Results
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If Absolute Error in x = Δx
Some definitions
Relative Error in x Δxx=
Percentage Error in xΔxx= × 100
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Q15. Using differentials, find the appropriate value of (3.968)3/2 .
(CBSE 2014 - 4 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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Q16. Using differentials, find approximate value of .
(CBSE Delhi 2012 - 4 marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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MEAN VALUE THEOREMS
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If a function f(x) satisfies
1) f(x) is continuous in [a ,b]
2) f(x) is differentiable in (a ,b)
then ∃ c ∈ (a , b) s.t.
LMVT-Lagrange mean value Theorem
f ′(c) =b – a
f(b) – f(a)
Becomes equal to instantaneous rate of change at some point inside the interval
Average rate of change over the interval
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LMVT-Lagrange mean value Theorem
Consider a chord joining A and B
c ∈(a,b)
B(b, f(b))
(a, f (a))
A
c f ′(c) =b – a
f(b) – f(a)
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Rolle’s Theorem
If y = f (x) is a function which is
1. Continuous in the interval [a, b]2. Differentiable in interval (a, b)
Then, there exists at least one c∈ (a , b) such that f ′(c) = 0.
A(a, f (a))
B(b, f(b))
C
3. f (a) = f (b)
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There may be more than one such value of ‘c’
B
Rolle’s theorem says atleastone ‘c’ where f ′(c) = 0
Example
Ac
2
c1
c3
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Q1. Verify Rolle's theorem for the function
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Q2. Verify Rolle's theorem for the following functions on indicated intervals,
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Q3. Verify Lagrange's mean value theorem for the following functions on the indicated intervals. Also, find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem:
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Q. Find the area of the greatest rectangle that can be inscribed in an ellipse
(CBSE 2013 - 6 Marks)
12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION
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SOLUTION
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SOLUTION
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SOLUTION
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8th Jan 2020-(Shift 1)
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8th Jan 2020-(Shift 1)
Thank You