(a) f(x) is a polynomial function, therefore the function is continuous on [1, 2].
(b) f'(x) = 1.(x - 2)2 + (x - 1).2(x - 2)
= (x - 2)[x - 2 + 2x - 2]
= (x - 2)(3x - 4)
Thus, function is differentiable on (1, 2)
(c) f(1) = 0 = f(2).
All the three conditions of Rolle’s theorem are true
at least one point c
(1, 2) s.t. f'(c) = 0
(c - 2)(3c - 4) = 0
c = 4/3 or c = 2
but 2 (1, 2), hence this choice is rejected and the value of c is 4/3.
E2= Student guesses the answer, and
A = answer is correct
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
X | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
P(X) | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 |
Two regression lines are represented by 2x + 3y – 20 = 0 and 4x + y – 15 = 0. Find the line of regression of y on x.
Find the equation of the two lines of regression for the following observations:
(3, 6), (4, 5), (5, 4),(6, 2),(7, 3)
Find an estimate of y for x = 3.5.
The constructed table is given below:
Given the total cost function for x units of a
commodity as
(i) the marginal cost function
(ii) average cost function
(iii) slope of average cost function
The fixed cost of a new product is `3600 and the variable cost per unit is ` 110. If the demand function is p(x) = 800 – 30x, find the breakeven values.
Let x units of the product be produced and sold.
Variable cost of producing x units = ` 110x
As the fixed cost is ` 3600,
So, total cost of producing x units,
C(x) = `(3600 + 110x)
Given demand function, p(x) = 800 – 30x
i.e., the selling price per unit = `(800 – 30x)
So, total revenue on selling x units,
R(x) = (price per unit)(number of units sold)
= `(800 – 30x)x
At breakeven values,
C(x) = R(x)
3600 + 110x = (800 – 30x)x
3600 + 110x = 800x – 30x2
30x2 – 690x + 3600 = 0
Dividing by 30, we get
x2 – 23 +120 = 0
(x – 8)(x – 15) = 0
This implies that either x = 8 or x = 15
Hence, the breakeven values of x are 8 and 15.
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