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प्रश्न
The slope of tangent at any point (a, b) is also called as ______.
उत्तर
The slope of tangent at any point (a, b) is also called as gradient.
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = | x | − | x − 1 | is monotonically increasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Function f(x) = loga x is increasing on R, if
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Show that f(x) = x – cos x is increasing for all x.
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = tan-1 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Function given by f(x) = sin x is strictly increasing in.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
