The least value of 'a' such that the function $f(x)=x^2+a x+1$ is increasing on (1, 2) is:
- A$0$
- ✓
- C
- D-4
Answer: B.
View full solution →Question types
Differentiation and it's Applications question types
83 questions across 5 question groups — pick any mix to generate a Applied Maths paper with step-by-step answer keys.
83
Questions
5
Question groups
5
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01
MCQ
29 Q→022 Marks Questions
22 Q→033 Marks Question
11 Q→045 Marks Questions
16 Q→05Case study (4 Marks)
5 Q→Sample Questions
Differentiation and it's Applications questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the total revenue (₹) received from the sale of x units of a product is given by: $R(x)=3 x^2+36 x+5$
- A₹ 116
- B₹ 96
- C₹ 90
- ✓₹ 126
Answer: D.
View full solution →If $y=A e^{5 x}+B e^{-5 x}$, then $\frac{d^2 y}{d x^2}$ is:
- ✓25y
- B5y
- C- 25y
- D15y
Answer: A.
View full solution →If C(x) and R(x) are respectively Cost function and Revenue function, then the Profit function P(x) is given by:
- AP(x) = R(x)
- BP(x) = C(x) + R(x)
- ✓P(x) = R(x) - C(x)
- DP(x) =R(x).C(x)
Answer: C.
View full solution →The point on the curve $x^2= 2 y$ which is nearest to the point (0, 5) is:
- ✓$(2 \sqrt{2}, 4)$
- B$(2 \sqrt{2}, 0)$
- C(0,0)
- D(2, 2)
Answer: A.
View full solution →
The total cost function (in thousands) for manufacturing x manipulators per year is given $C(x)=375+25 x-0.25 x^2 \quad 0 \leq x \leq 50$
(a) Use the marginal cost function to approximate the cost of producing the $31^{s t}$ manipulator.
(b) Use the total cost function to find the exact cost of producing the $31^{s t}$ manipulator.
View full solution →(a) Use the marginal cost function to approximate the cost of producing the $31^{s t}$ manipulator.
(b) Use the total cost function to find the exact cost of producing the $31^{s t}$ manipulator.
Find the interval in which the function f given by $f(x)=2 x^2-3 x$is (a) Increasing (b) Decreasing
View full solution →Suppose the total cost C(x) (in millions) for manufacturing x air-planes per year is given by the function
$C(x)=6+\sqrt{4 x+4} \quad 0 \leq x \leq 30$
a) Find the marginal cost at a production level of x air-planes per year.
(b) Find the marginal cost at a production level of 15 and 24 air-planes per year, and interpret the results.
View full solution →$C(x)=6+\sqrt{4 x+4} \quad 0 \leq x \leq 30$
a) Find the marginal cost at a production level of x air-planes per year.
(b) Find the marginal cost at a production level of 15 and 24 air-planes per year, and interpret the results.
A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 m and volume is $75 m^3$. If building of tank costs ₹ 100 per square metre for the base and ₹ 50 per square metre for the sides, find the cost of least expensive tank.
View full solution →A jet of enemy is flying along the curve $y=x^2+2$and a soldier is placed at the point (3, 2) . Find the minimum distance between the soldier and the jet.
View full solution →An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width.
Find the maximum value of $\left(\frac{1}{x}\right)^x$.
View full solution →Gymnast Clothing manufactures expensive hockey jerseys for sale to college bookstores in runs of up to $1 5 0$. Its cost (in rupees) for a run of $x$ hockey jerseys is
$C(x)=1500+10 x+0.2 x^2,(0 \leq x \leq 150)$
(a) Gymnast Clothing sells the jerseys at ₹ 90 eac. Find the revenue function
(b) Find the profit function
(c) How many should Gymnast Clothing manufacture to make a profit ? (ROund your answer up to the nearest whole number)
View full solution →$C(x)=1500+10 x+0.2 x^2,(0 \leq x \leq 150)$
(a) Gymnast Clothing sells the jerseys at ₹ 90 eac. Find the revenue function
(b) Find the profit function
(c) How many should Gymnast Clothing manufacture to make a profit ? (ROund your answer up to the nearest whole number)
The cost function for the manufacture of $x$ number of goods by a company is $C(x)=x^3-9 x^2+24 x$. Find the level of output at which the marginal cost is minimum. Further, if the selling price of a unit is $2 x^3+9 x^2$, find the average profit.
View full solution →Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3 \sqrt{3} cm$. (Leave the answer in terms of $\pi$ ).
View full solution →A closed right circular cylinder has volume $\frac{539}{2}$ cubic units. Find the radius and the height of the cylinder so that the total surface area is minimum.
View full solution →$A B C$ is a right angled triangle of given area $S$. Find the sides of the triangle for which the area of circumscribed circle is least.
View full solution →Read the following text and answer the following questions on the basis of the same:
A manufacturing company manufactures toys, the company observed the following costs at different production levels,
View full solution →A manufacturing company manufactures toys, the company observed the following costs at different production levels,
| Number of toys manufactured | Cost of raw material (₹) | Cost of Produc-tion (₹) | Cost of freight (₹) | Property tax (₹) | Salaries (₹) |
| 100 | 800 | 2000 | 1000 | 5000 | 20000 |
| 150 | 1200 | 3000 | 1500 | 5000 | 20000 |
| 200 | 1600 | 4000 | 2000 | 5000 | 20000 |
| 250 | 2000 | 5000 | 2500 | 5000 | 20000 |
| 300 | 2400 | 6000 | 3000 | 5000 | 20000 |
Q.1. Which of the following is the fixed cost
(A) Number of toys manufactured (B) Cost of raw material
(C) Cost of production supply (D) Salaries
Q.2. Total cost C(x) of toys for 'x' units of production is
(A) $C(x)=8 x^2+36 x+25000$
(B) $C(x)=8 x^2+30 x+20000$
(C) $C(x)=38 x+25000$
(D) $C(x)=28 x+25000$
Q.3. If the company observes the price p' per unit of item sold p = 5000 - 10x where x' is the number of units sold. Then the revenue function R(x) is given by.
(A) $R(x)=3000 x-10 x^2$
(B) $R(p)=5000 p-10 x^2$
(C) $R(x)=5000-10 x^2$
(D) $R(p)=5000-10 p^2$
Q.4. The marginal revenue (MR) of the company is given by
(A) 5000-20x
(B) 5000-20p
(C) - 20x
(D) - 20p
Read the following text and answer the following questions on the basis of the same:
A farmer has a piece of land. He observed that he got 600 units of fruits per tree by planting upto 25 trees and when 26 trees were grown, he received 15210 units of fruits, for 27 trees he ended up with 15390 fruits, for 28 trees he got 15540 fruits and this sequence of production of fruits continues in the same pattern as more trees, in excess of 25, were grown
Q.1. If 'x' more trees, in excess of 25 are grown, then the number of fruits produced per tree is
(A) 600-15x
(B) 600 + 15x
(C) 600x-15
(D) 600x + 15
Q.2. The product of entire garden if x' more trees, in excess of 25, are planted
(A) (25 + x)(600 + 15x)
(B) (25 - x)(600 - 15x)
(C) (25 + x)(600 - 15x)
(D) (25 + x)(15x - 600)
Q.3. The marginal production of the garden when 'x' more trees, in excess of 25, are
(A) 225 + 30x
(B) 225-30x
(C) 225x + 30
(D) 225x-30
Q.4. The critical point of producing 'x' more units of tree is
(A) 7 (B) 8 (C) 7.5 (D) 8.5
A farmer has a piece of land. He observed that he got 600 units of fruits per tree by planting upto 25 trees and when 26 trees were grown, he received 15210 units of fruits, for 27 trees he ended up with 15390 fruits, for 28 trees he got 15540 fruits and this sequence of production of fruits continues in the same pattern as more trees, in excess of 25, were grown
Q.1. If 'x' more trees, in excess of 25 are grown, then the number of fruits produced per tree is
(A) 600-15x
(B) 600 + 15x
(C) 600x-15
(D) 600x + 15
Q.2. The product of entire garden if x' more trees, in excess of 25, are planted
(A) (25 + x)(600 + 15x)
(B) (25 - x)(600 - 15x)
(C) (25 + x)(600 - 15x)
(D) (25 + x)(15x - 600)
Q.3. The marginal production of the garden when 'x' more trees, in excess of 25, are
(A) 225 + 30x
(B) 225-30x
(C) 225x + 30
(D) 225x-30
Q.4. The critical point of producing 'x' more units of tree is
(A) 7 (B) 8 (C) 7.5 (D) 8.5
Read the following text and answer the following questions on the basis of the same: "
A group of people decided to start a new project related to print media. They all have decided to publish a newspaper which has only college related news. They named their newspaper 'The Collegiate Investigator'. This newspaper has fixed production cost of ₹ 70 per edition and marginal printing and distribution cost of ₹ 0.40/per copy. The collegiate Investigator sells for ₹0.50/per copy.
Q.1. The associated cost function is:
(A) C(x) = 0.4x + 70
(B) C(x) = 0.4x - 70
(C) C(x) = - 0.4 + 70x
(D) C(x) = - 0.4 - 70x
Q.2. The associated revenue function is:
(A) R(x) = 0.4x
(B) R(x) = 0.5x
(C) R(x) = 0.7x
(D) R(x) = 0.5x + 70
Q.3. The associated profit function is:
(A) P(x) = - 0.1x - 70
(B) P(x) = - 0.1x + 50
(C) P(x) = 0.1x - 70
(D) P(x) = - 0.1x + 70
Q.4. What profit (or loss) results from the sale of 500 copies of The Collegiate Investigator?
(A) 20 loss "
(B) 20 profit
(C) 70 loss
(D) 70 profit
A group of people decided to start a new project related to print media. They all have decided to publish a newspaper which has only college related news. They named their newspaper 'The Collegiate Investigator'. This newspaper has fixed production cost of ₹ 70 per edition and marginal printing and distribution cost of ₹ 0.40/per copy. The collegiate Investigator sells for ₹0.50/per copy.
Q.1. The associated cost function is:
(A) C(x) = 0.4x + 70
(B) C(x) = 0.4x - 70
(C) C(x) = - 0.4 + 70x
(D) C(x) = - 0.4 - 70x
Q.2. The associated revenue function is:
(A) R(x) = 0.4x
(B) R(x) = 0.5x
(C) R(x) = 0.7x
(D) R(x) = 0.5x + 70
Q.3. The associated profit function is:
(A) P(x) = - 0.1x - 70
(B) P(x) = - 0.1x + 50
(C) P(x) = 0.1x - 70
(D) P(x) = - 0.1x + 70
Q.4. What profit (or loss) results from the sale of 500 copies of The Collegiate Investigator?
(A) 20 loss "
(B) 20 profit
(C) 70 loss
(D) 70 profit
Read the following text and answer the following questions on the basis of the same:
The market research department of a company recommends manufacture and market a new toy-car. The financial department provides the following cost function (in rupees) C(x) = 7,000 + $2 x ; 0 \leq x \leq 10,000$ where ₹7,000 is the estimate of fixed costs and 2 is the estimate of variable cost per toy-car. The estimate of revenue function (in rupees) is$R(x)=x(10-0.001 x) ; 0 \leq x \leq 10,000$.
Q.1. The marginal cost function is:
(A) 2x
(B) 2
(C) 7000+2x
(D) 7000
Q.2. The marginal revenue function is:
(A) 10-0.002x
(B) x(10-0.001x
(C) -0.002x
(D) 10
Q.3. The marginal revenue at x = 2000, 5000, 7000
(A) 6,-4,0
(B) 0,-4,6
(C) 6,0,-4
(D) none of these
Q.4. The profit function P(x) is given by
(A) $0.001 x^2+8 x+7000$
(B) $0.001 x^2-8 x-7000$
(C) $-0.001 x^2+8 x-7000$
(D) $-0.001 x^2-8 x-7000$
View full solution →The market research department of a company recommends manufacture and market a new toy-car. The financial department provides the following cost function (in rupees) C(x) = 7,000 + $2 x ; 0 \leq x \leq 10,000$ where ₹7,000 is the estimate of fixed costs and 2 is the estimate of variable cost per toy-car. The estimate of revenue function (in rupees) is$R(x)=x(10-0.001 x) ; 0 \leq x \leq 10,000$.
Q.1. The marginal cost function is:
(A) 2x
(B) 2
(C) 7000+2x
(D) 7000
Q.2. The marginal revenue function is:
(A) 10-0.002x
(B) x(10-0.001x
(C) -0.002x
(D) 10
Q.3. The marginal revenue at x = 2000, 5000, 7000
(A) 6,-4,0
(B) 0,-4,6
(C) 6,0,-4
(D) none of these
Q.4. The profit function P(x) is given by
(A) $0.001 x^2+8 x+7000$
(B) $0.001 x^2-8 x-7000$
(C) $-0.001 x^2+8 x-7000$
(D) $-0.001 x^2-8 x-7000$
Read the following text and answer the following questions on the basis of the same:
Four friends Rohan, Riya, Kartik and Mayank are playing number game. One by one they are giving the riddles to others based on the numbers. Now, its Riya's chance to give a number riddle. Riya said: "The sum of three positive numbers is 26. The second number is thrice as large as the first and sum of the squares of the number is least".
Q.1. If x, y, z are first, second and third number respectively, then which of the following is true.
(A) x + y + z = 26 and y = 3x
(B) x + y + z = 26 and x = 3y
(C) x + y + z = 26 and z = 3x
(D) x + y + z = 26 and y = 3z
Q.2. If S is the sum of the squares of the three numbers, then which of the following relation is true.
(A) $S=26 x^2+208 x+676$
(B) $S =26 x ^2-208 x -676$
(C) $5=26 x^2-208 x+675$
(D) $5=-26 x^2-208 x+676$
Q.3. The second derivative of S with respect to x is:
(A) 52
(B) 25
(C) 52x - 208
(D) 52x + 208
Q.4. At what values of x, the sum of the square of the numbers is least?
(A) 5
(B) 2
(C) 8
(D) 4
View full solution →Four friends Rohan, Riya, Kartik and Mayank are playing number game. One by one they are giving the riddles to others based on the numbers. Now, its Riya's chance to give a number riddle. Riya said: "The sum of three positive numbers is 26. The second number is thrice as large as the first and sum of the squares of the number is least".
Q.1. If x, y, z are first, second and third number respectively, then which of the following is true.
(A) x + y + z = 26 and y = 3x
(B) x + y + z = 26 and x = 3y
(C) x + y + z = 26 and z = 3x
(D) x + y + z = 26 and y = 3z
Q.2. If S is the sum of the squares of the three numbers, then which of the following relation is true.
(A) $S=26 x^2+208 x+676$
(B) $S =26 x ^2-208 x -676$
(C) $5=26 x^2-208 x+675$
(D) $5=-26 x^2-208 x+676$
Q.3. The second derivative of S with respect to x is:
(A) 52
(B) 25
(C) 52x - 208
(D) 52x + 208
Q.4. At what values of x, the sum of the square of the numbers is least?
(A) 5
(B) 2
(C) 8
(D) 4
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