Applications of Derivatives Exam Prep
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Applications of Derivatives Exam Prep
English 61 leveled MCQs Free
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Sample questions with model answers
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5 marks Long answer
Show that of all rectangles inscribed in a fixed circle, the square has the maximum area.
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The square inscribed in a circle has the maximum area due to symmetry and equal side lengths.
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5 marks Long answer
Find the interval in which 2x^3 + 2x^2 + 12x - 1 is strictly increasing.
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The function is strictly increasing where its derivative is positive.
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5 marks Long answer
Find positive number whose sum is 16 and the sum of whose cubes is minimum.
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The numbers are 8 and 8.
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5 marks Long answer
Find the intervals in which the function f(x)=2x^3−24x+25 is increasing or decreasing.
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Analyze the first derivative to determine intervals of increase and decrease.
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5 marks Long answer
Show that of all rectangles with a given area, the square has the least perimeter.
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The square minimizes perimeter for a given area due to equal side lengths.
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5 marks Long answer
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3.
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Height = 2R/√3.
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5 marks Long answer
An open tank with square base and vertical sides is to be constructed so as to hold a given quantity of water. Show that the cost of material is least when the depth of the tank is half the width.
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Depth = 0.5 * Width minimizes cost.
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5 marks Long answer
The combined resistance R of two resistors R1 and R2 is given by 1/R=1/R1+1/R2, R1,R2>0. If R1+R2=C, where C is a constant, show that the maximum resistance R is obtained by setting R1=R2.
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Max resistance occurs at R1=R2.
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5 marks Long answer
Show that among all rectangles inscribed in a given circle, the square has the maximum area.
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The square has the maximum area among inscribed rectangles.
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5 marks Long answer
Determine two positive integers whose sum is 15 and the product of the square of one with the other number is maximum.
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Integers are 7 and 8.
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5 marks Long answer
A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum?
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Lengths are 14m for square and 14m for circle.
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5 marks Long answer
Show that of all rectangles with given perimeter, the square has the maximum area.
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Square maximizes area for fixed perimeter.
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4 marks Medium answer
Find the intervals in which the function \( f(x) = 2x^2 - 3x \) is increasing.
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Increasing on \( (1.5, \infty) \)
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3 marks Medium answer
The radius of a circle is increasing uniformly at the rate of 3 cm/sec. Find the rate at which the area of the circle is increasing when the radius is 15 cm.
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Rate of area increase = 90π cm²/sec
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3 marks Medium answer
Let \( r, h, V \) be the radius, height and volume of a cone having semi-vertical angle \( heta \) with slant height \( l \), then show that \( V = rac{1}{3} \pi r^2 h \).
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Volume \( V = rac{1}{3} \pi r^2 h \)
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3 marks Medium answer
Show that the semi-vertical angle of the cone of maximum volume is \( an^{-1}(\sqrt{2}) \).
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Semi-vertical angle \( heta = an^{-1}(\sqrt{2}) \)
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3 marks Medium answer
Find the interval in which the function \( f(x) = x^2 - 4x + 6 \) is increasing or decreasing in \( R \).
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Increasing on \( (2, \infty) \); decreasing on \( (-\infty, 2) \)
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3 marks Medium answer
Consider two concentric circles \( C_1 \) and \( C_2 \) with radii \( r_1 \) and \( r_2 \) respectively, given \( r_1 < r_2 \). Given \( r_1 \) increases at the rate of 0.5 cm/s and \( r_2 \) decreases at the rate of 0.7 cm/s. Find the rate of change of the area between \( C_1 \) and \( C_2 \) when \( r_1 = 3 \) cm and \( r_2 = 5 \) cm.
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Rate of change of area = -1.2π cm²/s
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3 marks Medium answer
Find the intervals in which the function \( f(x) = x^3 - 3x + 1 \) is decreasing and increasing.
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Increasing on \( (-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty) \); decreasing on \( (-\sqrt{3}, \sqrt{3}) \)
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3 marks Medium answer
Find the intervals in which the function \( f(x) = x^2 - 4x + 6 \) is strictly increasing and strictly decreasing.
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Strictly increasing on \( (2, \infty) \); strictly decreasing on \( (-\infty, 2) \)
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3 marks Medium answer
Find the rate of change of area of a circle with respect to its radius \( r \) when \( r = 5 \).
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Rate of change = 10π cm²/s
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3 marks Medium answer
Find the local maxima or local minima of the function \( f(x) = x^3 - 4x + 1 \) if it exists.
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Local minima at \( x = 0 \)
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3 marks Medium answer
The radius of a circle is increasing uniformly at the rate of 5 cm/sec. Find the rate at which the area of the circle is increasing when the radius is 8 cm.
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Rate of area increase = 40π cm²/sec
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3 marks Medium answer
An edge of a cube is increasing at the rate of 4 cm/s. How fast is the volume increasing when the edge is 20 cm?
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Volume increasing at 480 cm³/s
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3 marks Medium answer
Find the interval in which the function \( f(x) = x^2 - 6x + 5 \) is increasing.
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Increasing on \( (3, \infty) \)
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3 marks Medium answer
The length \( x \) of a rectangle is increasing at the rate of 4 cm/s and the width \( y \) is decreasing at the rate of 5 cm/s. Find the rates of change of its area when \( x = 10 \) cm and \( y = 5 \) cm.
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Rate of change of area = -15 cm²/s
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3 marks Medium answer
Show that the function \( f(x) = x^3 + 3x + 5 \) is strictly increasing on \( R \).
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f'(x) > 0 for all \( x \in R \)
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3 marks Medium answer
Find the interval at which \( f(x) = 10 - 6x - 2x^2 \) is increasing or decreasing.
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Increasing on \( (-\infty, -3) \) and decreasing on \( (-3, \infty) \)
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3 marks Medium answer
Show that the function \( f(x) = x^3 - 3x^2 + 4x \) is always increasing on \( R \).
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f'(x) > 0 for all \( x \in R \)
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3 marks Medium answer
Find the minimum value of the function \( f(x) = x^2 + 1 \).
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Minimum value is 1 at \( x = 0 \)
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3 marks Medium answer
Find the rate of change of area of a circle with respect to its radius \( r \) when \( r = 5 \) cm.
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Rate of change = 10π cm²/s
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3 marks Medium answer
Show that the function \( f(x) = 2x^3 - 3x^2 + 4x \) is always increasing on \( R \).
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f'(x) > 0 for all \( x \in R \)
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3 marks Medium answer
Find the intervals in which the function f(x)=x^2−4x+6 is strictly increasing.
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The function is increasing where its derivative is positive.
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3 marks Medium answer
Find the minimum and maximum value, if any, of the function f(x)=(2x−1)^2+3.
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Minimum value is 3 at x=0.5.
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3 marks Medium answer
Find two positive numbers whose sum is 15 and sum of whose squares is minimum.
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The numbers are 7.5 and 7.5.
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3 marks Medium answer
The radius of a cylinder increases at a rate of 1cm/s and its height decreases at a rate of 1cm/s. Find the rate of change of its volume when the radius is 5cm and the height is 15cm.
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Use the volume formula V=πr^2h and apply the chain rule.
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3 marks Medium answer
Find the intervals in which the function f(x)=x^2−2x+5 is strictly increasing.
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The function is increasing for all x.
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3 marks Medium answer
A spherical bubble is decreasing in volume at the rate of 2 cm³/sec. Find the rate at which the surface area is diminishing when the radius is 3cm.
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Use the relationship between volume and surface area to find the rate.
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3 marks Medium answer
Find two numbers whose sum is 24 and whose product is as large as possible.
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The numbers are 12 and 12.
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3 marks Medium answer
Show that the semi-vertical angles of the cone of maximum volume and of given slant height is tan⁻¹(√2).
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Semi-vertical angle = tan⁻¹(√2).
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3 marks Medium answer
Show that of all rectangles of given perimeter, the square has the largest area.
Show answer outline
Square maximizes area for fixed perimeter.
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3 marks Medium answer
Find the least value of 'a' such that the function f(x)=x²+ax+1 is increasing on (1,2).
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Least value of a = -2.
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2 marks Short answer
What is the maximum value of the function \( f(x) = x^2 + 4x + 4 \)?
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Maximum value is 4 at \( x = -2 \)
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2 marks Short answer
A spherical bubble is decreasing in volume at the rate of 2 c.c/s. Find the rate at which the surface area is diminishing when the radius is 3 cm.
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Rate of surface area diminishing = 12π cm²/s.
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2 marks Short answer
Prove that the function f(x)=x³−3x²+3x−100 is increasing on R.
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f'(x) > 0 for all x.
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2 marks Short answer
The radius of a circle is increasing at the rate of 3 cm./sec. What rate is the area increasing when the radius is 10 cm?
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Rate of area increasing = 60π cm²/s.
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2 marks Short answer
Find the least value of 'a' such that f(x)=x²+ax+1 is increasing on [1,2].
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Least value of a = -2.
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2 marks Short answer
Show that f(x)=x³+x²+x+1 does not attain the maximum or minimum value for any real number.
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f(x) is unbounded as x → ±∞.
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2 marks Short answer
The radius of a circular plate is increasing at the rate of 0.2 cm/sec. At what rate is the area increasing when the radius of the plate is 2 cm?
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Rate of area increasing = 0.8π cm²/s.
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2 marks Short answer
If the radius of a circle is increased from 6 to 6.1 cm, find the approximate increase in its area.
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Approximate increase = 1.2π cm².
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1 marks Short answer
Find the intervals in which the function \( f(x) = x^3 - 4x \) is increasing.
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(0, 2)
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1 marks Short answer
Which of the following is an increasing function in \( R \)?
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f(x) = x^3
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1 marks Short answer
The figure shows the graph of \( f'(x) \). Then the function \( f(x) \) has a local maximum at \( x = 4 \).
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has a local maximum at x = 4
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1 marks Short answer
Which of the following function is increasing in its domain?
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f(x) = 2x^3 - 3x^2 + 4
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1 marks Short answer
Which of the following function is always increasing on \( R \)?
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f(x) = x^4
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1 marks Short answer
Maximum value of f(x)=e^x in [1,2] is
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e
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1 marks Short answer
The rate of change of volume of a sphere with respect to its radius when radius is 1 unit.
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4π
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1 marks Short answer
The rate of change of area of a circle with respect to radius r, when r = 5 cm.
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25π cm²/s
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1 marks Short answer
Which of the following functions has neither local maxima nor local minima?
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f(x)=logx
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1 marks Short answer
Which of the following functions is always increasing?
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2x+x^3
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