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QUESTION BANK mathematics (APPLICATION OF DERIVATIVES)
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THIS PDF CONTAINS QUESTION BANK OF CLASS 12TH CHAPTER APPLICATION OF DERIVATIVES ALONG WITH THE ANSWER KEY AT THE END
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Application of DerivativesQUICK RECAP
RATE OF CHANGE OF QUANTITIES INCREASING AND DECREASING
FUNCTIONS
8 Let y = f(x) be a function. If the change in
one quantity y varies with another quantity x, A function f is said to be
dy 8 an increasing function on an interval (a, b)
then or f ′(x) denotes the rate of change
dx if, for all x1, x2 ∈ (a, b), x2 > x1 ⇒ f(x2) ≥ f(x1)
of y (or f(x)) with respect to x. and strictly increasing if x2 > x1 ⇒ f(x2) > f(x1)
,8 a decreasing function on an interval (a, b) if, APPROXIMATIONS
for all x1, x2 ∈ (a, b), x2 > x1 ⇒ f(x2) ≤ f(x1) and
8 Let y = f(x) be any function of x. Let Dx be
strictly decreasing if x2 > x1 ⇒ f(x2) < f(x1) the change in x and Dy be the corresponding
Note : If f is a continuous function on [a, b] dy
change in y. Then, ∆y = . ∆x
and differentiable on (a, b), then for each dx
x ∈ (a, b) Note :
(i) f(x + Dx) ≈ f(x) + Dx f ′(x) is the
(i) f ′(x) > 0 ⇒ f is strictly increasing on (a, b)
approximate value.
(ii) f ′(x) < 0 ⇒ f is strictly decreasing on (a, b)
(ii) The change Dx is an error in x.
(iii) f ′(x) = 0 ⇒ f is a constant function on ∆x
(a, b). (iii) is the relative error in x.
x
TANGENTS AND NORMALS ∆x
(iv) × 100 is the percentage of error in x.
X Let y = f(x) be a curve and let x
P(x1, y1) be a point on it. Then,
dy MAXIMA AND MINIMA
= tan ψ = Slope of the tangent at P,
dx P 8 Let f be a real valued function defined on I
where y is the angle which the tangent at (subset of R), then
P(x1, y1) makes with the positive direction of X f is said to have a maximum value in
x-axis. I, if there exists a point c in I such that
f(c) > f(x) for all x ∈ I. The number f(c) is
X The equation of the tangent at P(x1, y1) to called the (absolute) maximum value of f in
dy
the curve y = f(x) is y − y1 = (x − x I and the point c is called point of maxima
dx P of f in I.
X The normal to a curve at P(x1, y1) is a line X f is said to have a minimum value in I, if there
perpendicular to the tangent at P and passing exists a point d in I such that f(d) < f(x)
for all x ∈ I. The number f(d) is called the
through P.
(absolute) minimum value of f in I and the
\ Slope of the normal at P point d is called point of minima of f in I.
1 1
=− =− Note : If f : I → R be a differentiable function
Slope of the tangent at P dy and c be any interior point of I, then f ′(c) = 0
dx P if f attains its absolute maximum (minimum)
X The equation of the normal at P(x1, y1) to the value at c.
1
curve y = f(x) is y − y1 = − (x − x1 ) 8 A point c is local maximum of a function f(x)
dy if there is an open interval I containing c such
dx P that f(x) < f (c) for all x ∈ I.
Note :
8 A point c is local minimum of a function f(x)
dy
(i) If = ∞, then the tangent at if there is an open interval I containing c such
dx P
that f(c) < f(x) for all x ∈ I.
P(x1, y1) is parallel to y-axis and its
Note : If a function f is either increasing or
equation is x = x1.
decreasing in an interval I, then f is said to be
dy a monotonic function.
(ii) If = 0 , then the tangent at P(x1, y1)
dx P 8 Critical Point : If f : I → R, then a point c ∈ I
is parallel to x-axis and its equation is is called the critical point of f, if either f ′(c) = 0
y = y1. or f is not differentiable at c.
,8 Methods of Finding Local Maxima and (ii) If f ′(c) = 0 and f ′′ (c) > 0, then x = c is a
Local Minima point of local minima.
X First Derivative Test : Let f(x) be a function (iii) If f ′(c) = 0 and f ′′ (c) = 0, then use the
defined on an open interval I and f(x) be first-derivative test.
continuous at a critical point c in I. Then,
8 Working Rule for Finding Absolute
(i) If f ′(x) changes sign from positive to Maxima and Minima
negative as x increases through c, then
(i) Find all critical points of f in the interval
x = c is a point of local maxima.
I = [a, b] i.e., find all points x where either
(ii) If f ′(x) changes sign from negative to f ′(x) = 0 or f is not differentiable.
positive as x increases through c, then
(ii) Find the end points of the interval i.e.,
x = c is a point of local minima.
a and b.
(iii) If f ′(x) does not change sign as x
(iii) Find the values of f at all its critical points
increases through c, we say that c is
in interval I and at the end points of the
neither a point of local maxima nor a
interval I.
point of local minima. In this case, x = c
is called a point of inflection. (iv) Absolute maximum value of f in
X Second Derivative Test : Let f(x) be a I = greatest of values of f at end points
function defined on an interval I and c ∈ I. and at all critical points.
Let f be twice differentiable at c. Then, (v) Absolute minimum value of f in I = least
(i) If f ′(c) = 0 and f ′′ (c) < 0, then x = c is a of values of f at end points and at all
point of local maxima. critical points.
, Previous Years’ CBSE
PREVIOUS Board
YEARS MCQS Questions
6.2 Rate of Change of Quantities What value is reflected in this question?
(AI 2013C)
VSA (1 mark)
SA (2 marks)
1. The radius of a circle is increasing at the
uniform rate of 3 cm/sec. At the instant 6. The total cost C(x) associated with the
when the radius of the circle is 2 cm, its area production of x units of an item is given
increases at the rate of cm2/s. by C(x) = 0.005x3 – 0.02x2 + 30x + 5000.
(2020) Find the marginal cost when 3 units are
2. The amount of pollution content added in air produced, where by marginal cost we mean
in a city due to x-diesel vehicles is given by the instantaneous rate of change of total cost
P(x) = 0.005x3 + 0.02x2 + 30x. Find the at any level of output. (2018)
marginal increase in pollution content when 7. The volume of a sphere is increasing at the
3 diesel vehicles are added and write which rate of 3 cubic centimeter per second. Find
value is indicated in the above question. the rate of increase of its surface area, when
(Delhi 2013) the radius is 2 cm. (Delhi 2017)
3. The money to be spent for the welfare of the 8. The volume of a cube is increasing at the
employees of a firm is proportional to the rate of 9 cm3/s. How fast is its surface area
rate of change of its total revenue (marginal increasing when the length of an edge is 10 cm?
revenue). If the total revenue (in rupees) (AI 2017)
received from the sale of x units of a product
is given by R(x) = 3x2 + 36x + 5, find the LA 1 (4 marks)
marginal revenue, when x = 5, and write 9. A ladder 13 m long is leaning against a
which value does the question indicate. vertical wall. The bottom of the ladder is
(AI 2013) dragged away from the wall along the ground
4. The total cost C(x) associated with provision at the rate of 2 cm/sec. How fast is the height
of free mid-day meals to x students of a on the wall decreasing when the foot of the
school in primary classes is given by ladder is 5 m away from the wall? (AI 2019)
C(x) = 0.005x3 – 0.02x2 + 30x + 50 10. The side of an equilateral triangle is increasing
If the marginal cost is given by rate of change at the rate of 2 cm/s. At what rate is its area
dC increasing when the side of the triangle is
of total cost, write the marginal cost of 20 cm? (Delhi 2015)
dx
food for 300 students. What value is shown 11. The sides of an equilateral triangle are
here? (Delhi 2013C) increasing at the rate of 2 cm/sec. Find the
5. The total expenditure (in `) required for rate at which the area increases, when the
providing the cheap edition of a book for side is 10 cm. (AI 2014C)
poor and deserving students is given by 12. A ladder 5 m long is leaning against a wall.
R(x) = 3x2 + 36x, where x is the number of The bottom of the ladder is pulled along
sets of books. If the marginal expenditure the ground, away from the wall, at the rate
dR of 2 cm/s. How fast is its height on the wall
is defined as , write the marginal
dx decreasing when the foot of the ladder is
expenditure required for 1200 such sets. 4 m away from the wall? (AI 2012)
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