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महाराष्ट्र राज्य शिक्षण मंडळएचएससी वाणिज्य (इंग्रजी माध्यम) इयत्ता १२ वी

The slope of tangent at any point (a, b) is also called as ______. - Mathematics and Statistics

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प्रश्न

The slope of tangent at any point (a, b) is also called as ______.

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उत्तर

The slope of tangent at any point (a, b) is also called as gradient.

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पाठ 1.4: Applications of Derivatives - Q.2

व्हिडिओ ट्यूटोरियलVIEW ALL [3]

संबंधित प्रश्‍न

Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).


Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 107  ?


Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?


Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?


Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?


Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?


Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?


State whether f(x) = tan x − x is increasing or decreasing its domain ?


The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:


Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)


If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then


Find `dy/dx,if e^x+e^y=e^(x-y)`


Show that f(x) = x – cos x is increasing for all x.


Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing


By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.

Solution: f(x) = 2x3 – 15x2 – 84x – 7

∴ f'(x) = `square`

∴ f'(x) = 6`(square) (square)`

Since f(x) is decreasing function.

∴ f'(x) < 0

Case 1: `(square)` > 0 and (x + 2) < 0

∴ x ∈ `square`

Case 2: `(square)` < 0 and (x + 2) > 0

∴ x ∈ `square`

∴ f(x) is decreasing function if and only if x ∈ `square`


Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______


For every value of x, the function f(x) = `1/7^x` is ______ 


Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.


The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.


Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.


The values of a for which the function f(x) = sinx – ax + b increases on R are ______.


In case of decreasing functions, slope of tangent and hence derivative is ____________.


The function f (x) = x2, for all real x, is ____________.


Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.


y = log x satisfies for x > 1, the inequality ______.


Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.


Read the following passage:

The use of electric vehicles will curb air pollution in the long run.

The use of electric vehicles is increasing every year and the estimated electric vehicles in use at any time t is given by the function V:

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:

  1. Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
  2. Prove that the function V(t) is an increasing function. (2)

The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.


The function f(x) = x3 + 3x is increasing in interval ______.


Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.


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