Please select a subject first
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Choose the correct alternative:
Slope of the normal to the curve 2x2 + 3y2 = 5 at the point (1, 1) on it is
Concept: Introduction of Derivatives
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
Concept: Increasing and Decreasing Functions
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
Concept: Increasing and Decreasing Functions
The slope of tangent at any point (a, b) is also called as ______.
Concept: Increasing and Decreasing Functions
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Concept: Increasing and Decreasing Functions
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
Concept: Increasing and Decreasing Functions
State whether the following statement is True or False:
The equation of tangent to the curve y = x2 + 4x + 1 at (– 1, – 2) is 2x – y = 0
Concept: Introduction of Derivatives
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
Concept: Increasing and Decreasing Functions
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
Concept: Increasing and Decreasing Functions
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Concept: Maxima and Minima
The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 − 𝑥). Find x for which profit is increasing
Concept: Application of Derivatives to Economics
A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which profit is increasing
Solution: Total cost C = 40 + 2x and Price p = 120 − x
Profit π = R – C
∴ π = `square`
Differentiating w.r.t. x,
`("d"pi)/("d"x)` = `square`
Since Profit is increasing,
`("d"pi)/("d"x)` > 0
∴ Profit is increasing for `square`
Concept: Application of Derivatives to Economics
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Concept: Increasing and Decreasing Functions
Divide 20 into two ports, so that their product is maximum.
Concept: Maxima and Minima
Complete the following activity to find MPC, MPS, APC and APS, if the expenditure Ec of a person with income I is given as:
Ec = (0.0003)I2 + (0.075)I2
when I = 1000
Concept: Application of Derivatives to Economics
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
Concept: Maxima and Minima
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Concept: Methods of Integration: Integration by Parts
Choose the correct alternative:
`int(("e"^(2x) + "e"^(-2x))/"e"^x) "d"x` =
Concept: Integration
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
Concept: Methods of Integration: Integration Using Partial Fractions
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Concept: Methods of Integration: Integration by Parts
