If $f(x) = \log(1 + x) + \frac{1}{1+x}$, show that $f(x)$ attains its minimum value at $x = 0$.
Given: $f(x) = \log(1 + x) + \frac{1}{1+x}$
To show: $f(x) \text{ attains its minimum value at } x = 0$
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Applications of Derivatives - ISC Class 12 Mathematics Questions with Answers, Page 2
87 past-paper questions on Applications of Derivatives from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 21-40 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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A cylindrical popcorn tub of radius $10\text{ cm}$ is being filled with popcorns at the rate of $314\text{ cm}^3$ per minute. The level of the popcorns in the tub is increasing at the rate of:
- (a)$1\text{ cm/minute}$
- (b)$0\cdot 1\text{ cm/minute}$
- (c)$1\cdot 1\text{ cm/minute}$
- (d)$0\cdot 5\text{ cm/minute}$
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$y = x^5 + x^3, x \in \mathbb{R}$ is a function. Rina, Abha and Saurabh have given their opinions about the function in the following statements:
• Statement 1: Rina says that 'y' is an increasing function for all values of $x$.
• Statement 2: Abha says that 'y' is an odd function.
• Statement 3: Saurabh says that 'y' is symmetrical about the origin.
Related to the above statements, which of the following option is true?
- (a)Rina, Abha and Saurabh are correct.
- (b)Rina and Abha are correct, but Saurabh is wrong.
- (c)Rina and Saurabh are correct, but Abha is wrong.
- (d)Saurabh and Abha are correct, but Rina is wrong.
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Find the equation to the tangent at $(0,0)$ on the curve $y = 4x^2 - 2x^3$. [Analysis]
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A point $x = c$ is called the critical point of a function if:
- (a)$f'(c) = 0$
- (b)$f$ is not differentiable at $x = c$.
- (c)both (a) & (b).
- (d)none of the above.
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A cone of maximum volume is inscribed in a given sphere. Then prove that ratio of the height of the cone to the diameter of the sphere is equal to $\frac{2}{3}$. [Application]
To show: $\text{Ratio of height of cone to diameter of sphere} = \frac{2}{3}$
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The graph of $f(x) = -x^3 + 27x - 2$ is given below:
(a)[1.0]
Find the slope of the above graph.
(b)[2.0]
Find the co-ordinates of turning points, A and B.
(c)[1.0]
Evaluate $f''(-2)$, $f(0)$ and $f'(3)$ and arrange them in ascending order.
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Case Study:
The length of the perimeter of a slice of a pizza in the form of a sector of circle is 20 cm.
$r$ be the radius of the circle, sectorial angle be $\theta$ radian and $l$ be the length of the arc.
Based on the above information, answer the following questions.
(a)
Express the radius of the sector in terms of sectorial angle $\theta$ radian.
(b)
Let A be the area of the slice. Then, express A in terms of r.
(c)
For the maximum value of A, find the value of the sectorial angle.
(d)
Maximum area of the slice of the pizza is ____.
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Find the acute angle between the curves $y = |x^2 - 1|$ and $y = |x^2 - 3|$ at their point of intersection when $x > 0$.
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Let $f(x) = \frac{\log 5x}{kx}$, where $x > 0, k \in \mathbb{R}^+$.
(a)
Show that $f'(x) = \frac{1-\log 5x}{kx^2}$.
(b)
The graph of $f$ has exactly one maximum point at P. Find the $x$-coordinate of P.
(c)
Find $f''(x)$.
(d)
Find the value of $x$ for which $f''(x)$ vanishes.
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Assertion: $f(x)$ is a quadratic function.
Reason: Derivative of every quadratic function is linear.
- (a)Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
- (b)Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
- (c)Assertion is true and Reason is false.
- (d)Assertion is false and Reason is true.
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Given $2f(x) = \log|x|^a - bx^2 + x$, if $f(x)$ has extreme values at $x = -1$ and $x = 2$, find the value of $a$ and $b$.
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The displacement, $x\text{ m}$ of a particle from a fixed point at time '$t$' seconds is given by $x = 6\cos\left(3t + \frac{\pi}{3}\right)$.
Find the acceleration of the particle, when $t = \frac{2\pi}{3}$.
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Find the point on the curve $y = 2x^2 - 6x - 4$ at which the tangent is parallel to the $x$-axis.
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Given $f(x) = 2\log(x-2) - x^2 + 4x + 1$ and $f'(x) = \frac{-k(x-p)(x-q)}{(x-k)}$.
(a)
Find $k + p + q$.
(b)
The Statement “$f(x)$ is strictly increasing in $(-\infty, 1] \cup (2, 3]$” is false. Justify.
(c)
Hence, find the interval(s) in which the function is strictly increasing.
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In a particular forest, the population density ($P$) of wild animals (in thousands of animals per sq. km.) at a distance '$r$' km from the centre of the forest is approximately given by $P = 10 + 20r - 10r^2$
(a)[1.0]
What is the population density in the centre of the forest?
(b)[1.0]
Calculate the rate of change of population density.
(c)[1.0]
For what value of '$r$' is the population density maximum?
(d)[1.0]
Prove that the population density of animals varies with distance from the centre of the forest.
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Sonia watches a painting which has its bottom edge 2 meters (m) above eye level and its top edge is 3 m above eye level as shown in the diagram.
Based on the above information answer the questions that follow.
(a)
Given $\alpha$ and $\theta$ as shown in the diagram, find $\tan\alpha$ and $\tan(\alpha+\theta)$.
(b)
Find $\theta$ in terms of $x$ only.
(c)
Find $\frac{d\theta}{dx}$.
(d)
Find $x$ so that $\frac{d\theta}{dx} = 0$.
(e)
Use 1st derivative test, find the distance Sonia should stand from the wall to maximize her viewing angle of the painting.
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Find the equation of the normal to the curve $y = x^2 - 3x + 1$ at the point $(3, 1)$.
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Assertion: Maximum value of the function is 0.
Reason: Minimum value of the function approaches $\infty$.
Shown below the graph of $f(x) = -2|x-3|$.
Assertion (A): Maximum value of the function is 0.
Reason (R): Minimum value of the function approaches $\infty$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
$y = \ln(x+1) - \ln x$ is a curve. The tangent to the curve at the point $P(1, \ln 2)$ meets x-axis at A and y-axis at B. The normal to the curve at P meets the x-axis at C and y-axis at D.
(a)
Find the slope of tangent at P and find the slope of normal at P.
(b)
Find the equation of tangent at P.
(c)
Find the equation of normal at P.
(d)
Find the co-ordinates of A and C in terms of $\ln 2$.
No answer yet.