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Published on: 28/08/2019
Application of Derivatives
Download CBSE Class 12th Standard CBSE Maths question papers, sample papers, important questions, and previous year solved papers in PDF format. Get free study materials, NCERT solutions, and exam preparation resources for Class 12th Standard CBSE Maths
Questions + Answers key
Take MCQ Maths Test
1.
At what point on the curve y = x2 does the tangent make an angle of 450 with x-axis?
2.
Prove that the tangents on the curve y=x3+6 at the points (-1,5) and (1,7) are parallel.
3.
Find the least value of a such that the function f(x)=x+ax+1 is strictly increasing on(1,2).
4.
Show that the function f given by f(x) = x3-3x2+3x-100, X \(\in\) R is increasing on R.
5.
Show that f(x) = (x-1) ex+1 is an increasing function for x > 0.
6.
Show that,the function f(x) = log(cos x) is decreasing, in \([0,{\pi\over2}]\).
7.
Show that the function y = \({3\over x}+7\) is strictly decreasing for x \(\in\)R (x \(\neq \) 0).
8.
Prove that,the function f(x)=x3-3x2+3x+107 is increasing on R.
9.
The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x3+0.02x2 +30x.Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?
10.
Find the point on the curve y=x2-4x+5,where tangent to the curve is parallel to the x-axis.
1.
Slope of the tangent = tan 45° = 1.
y′ = 2x ⇒ 2x = 1
⇒ x = \(\frac{1}{2}\). Substituting in curve, we get y = \(\frac{1}{4}\) .
Points is (\(\frac{1}{2}\), \(\frac{1}{4}\)).
2.
y = x3 + 6 ⇒ y′ = 3x2
\(\left.\left.y^{\prime}\right]_{(-1,5)}=3(-1)^2=3 \text { and } y^{\prime}\right]_{(1,7)}=3(1)^2=3\)
As slope at these points are equal,hence tangents are parallel.
3.
a=-2
4.
Consider function f(x) = x3 - 3x2 + 6x - 100
f'(x) 3x2- 6x + 6 = 3(x2 - 2x + 2)
3[(x 1)2+1] > 0,for all x ∈ R
Hence, f is increasing on R.
5.
f′(x) = (x−1) ex + ex = x⋅ex, \(e^x>0 \forall x \text { and } x>0\)
∴ f′(x) > 0, for x > 0
Hence, function is increasing.
6.
\(f^{\prime}(x)=\frac{1}{\cos x} \cdot(-\sin x)=-\tan x, \tan x>0 \text { for }\left(0, \frac{\pi}{2}\right)\)
f′(x) < 0. Hence, function is strictly decreasing
7.
\(y^{\prime}=-\frac{3}{x^2}<0, \text { for } x \in R, x \neq 0\). Hence, function is strictly decreasing
8.
f'(x) 3x2- 6x + 3 = 3(x2 - 2x + 1)
= 3 (x - 1)2 > 0
Hence, function is increasing in R.
9.
30.255 Concern for environment;Responsibility for pollution free envirnment
10.
( )
\(\frac{d y}{d x}=2 x-4 \text { . For tangent to be parallel to the } x \text { -axis, }\frac{d y}{d x}=0\)
\(\Rightarrow 2 x-4=0 \Rightarrow x=2 \text { . Substituting in curve, we get }\)
\(y=4-8+5=1\)
Point is (2,1)
12th Standard CBSE Syllabus & Materials
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