Applications of Derivatives Chapter 6
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Questions and Answers

What is the marginal revenue when x = 15?

126

In which intervals is the function f(x) = 4x^3 - 6x^2 - 72x + 30 increasing?

  • (-2, 3)
  • (2, ∞)
  • (-∞, -2) and (2, ∞)
  • (-∞, -2) and (3, ∞) (correct)

    • In which intervals is the function f(x) = x^2 – 4x + 6 decreasing?

  • (-2, 3)
  • (3, ∞)
  • (-∞, 2) (correct)
  • (2, ∞)

    • The function f(x) = 7x - 3 is increasing on the real numbers.

    <p>True</p> Signup and view all the answers

    The function f(x) = sin 3x is _____ in the interval (π/4, 5π/4).

    <p>decreasing</p> Signup and view all the answers

    What does the derivative represent when one quantity varies with another quantity x, satisfying the rule y = f(x)?

    <p>Rate of change of y with respect to x</p> Signup and view all the answers

    What is the rate of change of the area of a circle with respect to its radius r when r = 5 cm?

    <p>10π cm2/s</p> Signup and view all the answers

    What is the rate of increase of the circumference of a circle if the radius is increasing at 0.7 cm/s?

    <p>1.4π cm/s</p> Signup and view all the answers

    The volume of a cube is increasing at 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm? The rate of surface area increase is __ cm2/s.

    <p>36</p> Signup and view all the answers

    Find the rate at which the volume of a balloon increases when its radius is 15 cm. The volume is increasing at:

    <p>300π cm3/s</p> Signup and view all the answers

    Marginal revenue is the rate of change of total revenue with respect to the number of items sold.

    <p>True</p> Signup and view all the answers

    Find all points of local maxima and local minima of the function f given by f(x) = x^3 - 3x + 3.

    <p>Critical points are x = ±1. Local minimum at x = 1 with a local minimum value f(1) = 1. Local maximum at x = -1 with a maximum value f(-1) = 5.</p> Signup and view all the answers

    What is the Second Derivative Test?

    <p>If f'(c) = 0 and f''(c) &lt; 0, then x = c is a point of local maxima. If f'(c) = 0 and f''(c) &gt; 0, then x = c is a point of local minima. If f'(c) = 0 and f''(c) = 0, the test fails.</p> Signup and view all the answers

    What is the function f(x) given in Example 34?

    <p>tan^(-1)(sin x + cos x)</p> Signup and view all the answers

    What is the derivative f'(x) of the function f(x) = tan^(-1)(sin x + cos x)?

    <p>(cos x - sin x) / (1 + (sin x + cos x)^2)</p> Signup and view all the answers

    Find the local minimum value of the function f given by f(x) = 3 + |x|, x ∈ R.

    <p>The local minimum value of f is 3, which occurs at x = 0.</p> Signup and view all the answers

    Find the local maximum and local minimum values of the function f given by f(x) = 3x^4 + 4x^3 - 12x^2 + 12.

    <p>Local maximum at x = 0 with a maximum value of f(0) = 12. Local minima at x = 1 with f(1) = 7 and x = -2 with f(-2) = -20.</p> Signup and view all the answers

    In which interval is the function f(x) = tan^(-1)(sin x + cos x) always increasing?

    <p>(0, 4/π)</p> Signup and view all the answers

    Find all points of local maxima and local minima of the function f given by f(x) = 2x^3 - 6x^2 + 6x + 5.

    <p>No points of local maxima or minima. x = 1 is a point of inflexion.</p> Signup and view all the answers

    What is the rate at which the area of the circular disc is increasing when the radius is 3.2 cm?

    <p>0.320π cm^2/s</p> Signup and view all the answers

    What is the volume of the largest box that can be constructed from a 3m by 8m rectangular sheet of aluminum?

    <p>200/27 m^3</p> Signup and view all the answers

    A cylindrical tank of radius 10m is being filled with wheat at the rate of 314 cubic meters per hour. What is the rate at which the depth of the wheat is increasing?

    <p>1 m/h</p> Signup and view all the answers

    On which of the following intervals is the function f given by $f(x) = x^{100} + ext{sin}x - 1$ decreasing?

    <p>(0, $rac{\pi}{2}$)</p> Signup and view all the answers

    For what values of $a$ is the function f given by $f(x) = x^2 + ax + 1$ increasing on [1, 2]?

    <p>a &lt; -3</p> Signup and view all the answers

    Prove that the function f given by $f(x) = \log(\text{sin}x)$ is increasing on ($0, \frac{\pi}{2}$) and decreasing on ($\frac{\pi}{2}$, $rac{\pi}{2}$)

    <p>The derivative of $f(x)$, $f'(x) = \frac{\text{cos}x}{\text{sin}x}$, is positive in ($0, \frac{\pi}{2}$) and negative in ($\frac{\pi}{2}$, $rac{\pi}{2}$), hence proving the claim.</p> Signup and view all the answers

    Prove that the function f given by $f(x) = \log|\text{cos}x|$ is decreasing on ($0, \frac{\pi}{2}$) and increasing on ($\frac{3\pi}{2}, 2\pi$)

    <p>The derivative of $f(x)$ is $f'(x) = -\frac{\text{sin}x}{|\text{cos}x|}$, which is negative in ($0, \frac{\pi}{2}$) and positive in ($\frac{3\pi}{2}, 2\pi$), thus proving the claim.</p> Signup and view all the answers

    Prove that the function given by $f(x) = x^3 - 3x^2 + 3x - 100$ is increasing in $R$

    <p>By finding the derivative $f'(x) = 3x^2 - 6x + 3$, which is always positive for all real $x$, we can conclude that the function is increasing in $R$.</p> Signup and view all the answers

    The interval in which $y = x^2e^{-x}$ is increasing is

    <p>(0, 2)</p> Signup and view all the answers

    Find the maximum and minimum values, if any, of f(x) = |x + 2| - 1.

    <p>Maximum value: 1, Minimum value: -3</p> Signup and view all the answers

    Find the maximum and minimum values, if any, of g(x) = -|x + 1| + 3.

    <p>Maximum value: 2, Minimum value: -4</p> Signup and view all the answers

    Find the maximum and minimum values, if any, of h(x) = sin(2x) + 5.

    <p>Maximum value: 6, Minimum value: 4</p> Signup and view all the answers

    Find the maximum and minimum values, if any, of f(x) = |sin(4x) + 3|.

    <p>Maximum value: 4, Minimum value: 2</p> Signup and view all the answers

    Find the maximum and minimum values, if any, of h(x) = x + 1 in the interval (-1, 1).

    <p>Maximum value: 2, Minimum value: 0</p> Signup and view all the answers

    Find the local maxima and local minima, if any, of f(x) = x^2.

    <p>Local maximum: None, Local minimum: 0</p> Signup and view all the answers

    Find the local maxima and local minima, if any, of g(x) = x^3 - 3x.

    <p>Local maximum: None, Local minimum: 1</p> Signup and view all the answers

    Find the local maxima and local minima, if any, of h(x) = sin(x) + cos(x) in the interval (0, π/2).

    <p>Local maximum: 2, Local minimum: 0</p> Signup and view all the answers

    Find the local maxima and local minima, if any, of f(x) = x^3 - 6x^2 + 9x + 15.

    <p>Local maximum: None, Local minimum: 3</p> Signup and view all the answers

    Find the local maxima and local minima, if any, of g(x) = 1/(2x), where x > 0.

    <p>Local maximum: None, Local minimum: 1</p> Signup and view all the answers

    Study Notes

    Application of Derivatives

    Rate of Change of Quantities

    • The derivative of a function represents the rate of change of the function with respect to its variable.
    • If y varies with x, then dy/dx represents the rate of change of y with respect to x.
    • If two variables x and y are varying with respect to another variable t, then dy/dx can be calculated using the chain rule.

    Examples of Rate of Change of Quantities

    • The rate of change of the area of a circle with respect to its radius r is given by dA/dr = 2πr.
    • The rate of change of the volume of a cube with respect to its edge length x is given by dV/dx = 3x^2.
    • The rate of change of the surface area of a cube with respect to its edge length x is given by dS/dx = 6x.

    Increasing and Decreasing Functions

    • A function is said to be increasing on an interval if the function values increase as the input values increase.
    • A function is said to be decreasing on an interval if the function values decrease as the input values increase.
    • A function is said to be constant on an interval if the function values remain the same as the input values change.

    First Derivative Test

    • If f'(x) > 0 for all x in an interval, then f is increasing on that interval.
    • If f'(x) < 0 for all x in an interval, then f is decreasing on that interval.
    • If f'(x) = 0 for all x in an interval, then f is constant on that interval.

    Examples of Increasing and Decreasing Functions

    • The function f(x) = x^2 is increasing on the interval [0, ∞) and decreasing on the interval (-∞, 0].
    • The function f(x) = 7x - 3 is increasing on the entire real line.
    • The function f(x) = cos x is decreasing on the interval (0, π) and increasing on the interval (π, 2π).

    Intervals of Increase and Decrease

    • The function f(x) = x^2 - 4x + 6 is decreasing on the interval (-∞, 2) and increasing on the interval (2, ∞).
    • The function f(x) = 4x^3 - 6x^2 - 72x + 30 is increasing on the intervals (-∞, -2) and (3, ∞), and decreasing on the interval (-2, 3).
    • The function f(x) = sin 3x is increasing on the intervals (0, π/6) and (π/2, 5π/6), and decreasing on the intervals (π/6, π/2) and (5π/6, π).Here are the study notes for the text:

    Increasing and Decreasing Functions

    • A function f is increasing in an interval I if for every x1, x2 ∈ I, x1 < x2 implies f(x1) ≤ f(x2)
    • A function f is decreasing in an interval I if for every x1, x2 ∈ I, x1 < x2 implies f(x1) ≥ f(x2)
    • A function f is strictly increasing in an interval I if for every x1, x2 ∈ I, x1 < x2 implies f(x1) < f(x2)
    • A function f is strictly decreasing in an interval I if for every x1, x2 ∈ I, x1 < x2 implies f(x1) > f(x2)

    Maximum and Minimum Values

    • A function f has a maximum value in an interval I if there exists a point c in I such that f(c) ≥ f(x) for all x ∈ I
    • A function f has a minimum value in an interval I if there exists a point c in I such that f(c) ≤ f(x) for all x ∈ I
    • An extreme value of a function f is either a maximum value or a minimum value of f

    Local Maxima and Minima

    • A point c is a point of local maxima of a function f if there is an h > 0 such that f(c) ≥ f(x) for all x in (c - h, c + h), x ≠ c
    • A point c is a point of local minima of a function f if there is an h > 0 such that f(c) ≤ f(x) for all x in (c - h, c + h)
    • The value f(c) is called the local maximum value or local minimum value of f

    First Derivative Test

    • If f'(x) changes sign from positive to negative as x increases through c, then c is a point of local maxima
    • If f'(x) changes sign from negative to positive as x increases through c, then c is a point of local minima
    • If f'(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima, and is called a point of inflection

    Second Derivative Test

    • If f'(c) = 0 and f"(c) < 0, then c is a point of local maxima
    • If f'(c) = 0 and f"(c) > 0, then c is a point of local minima
    • If f'(c) = 0 and f"(c) = 0, then the test fails, and we need to use the first derivative test

    Examples

    • The function f(x) = sin 3x has a local maximum value at x = π/6 and a local minimum value at x = π/2
    • The function f(x) = x2 has a local minimum value at x = 0
    • The function f(x) = 3 + |x| has a local minimum value at x = 0
    • The function f(x) = 3x4 + 4x3 - 12x2 + 12 has a local maximum value and a local minimum value, which can be found using the first derivative test### Application of Derivatives

    Example 21

    • Find the local maxima and minima of the function f(x) = 2x³ - 6x² + 6x + 5
    • f'(x) = 6x² - 12x + 6 = 6(x - 1)²
    • f''(x) = 12(x - 1)
    • f'(x) = 0 gives x = 1
    • f''(1) = 0, so second derivative test fails
    • Use first derivative test to find that x = 1 is a point of inflexion

    Example 22

    • Find two positive numbers whose sum is 15 and the sum of whose squares is minimum
    • Let one of the numbers be x, then the other number is 15 - x
    • S(x) = x² + (15 - x)² = 2x² - 30x + 225
    • S'(x) = 4x - 30
    • S'(x) = 0 gives x = 15/2
    • S''(x) = 4 > 0, so x = 15/2 is a point of local minima
    • The two numbers are 15/2 and 15/2

    Example 23

    • Find the shortest distance of the point (0, c) from the parabola y = x², where 1/2 ≤ c ≤ 5
    • Let (h, k) be a point on the parabola, then D = (h - 0)² + (k - c)² = h² + (k - c)²
    • D = k + (k - c)²
    • D'(k) = 1 + 2(k - c)
    • D'(k) = 0 gives k = (2c - 1)/2
    • D'(k) is minimum at k = (2c - 1)/2
    • The shortest distance is given by D = (2c - 1)/2 + ((2c - 1)/2 - c)²

    Example 24

    • Let AP and BQ be two vertical poles at points A and B, respectively
    • RP² + RQ² is minimum when AP = 16, BQ = 22, and AB = 20
    • Let AR = x, then RB = 20 - x
    • RP² = AR² + AP² = x² + 16²
    • RQ² = RB² + BQ² = (20 - x)² + 22²
    • S(x) = RP² + RQ² = 2x² - 40x + 1140
    • S'(x) = 4x - 40
    • S'(x) = 0 gives x = 10
    • S''(x) = 4 > 0, so x = 10 is a point of local minima
    • The distance of R from A is AR = x = 10

    Example 25

    • Find the area of the trapezium when it is maximum
    • The required trapezium has AP = x, QB = x, and DP = QC = √(100 - x²)
    • A = (sum of parallel sides)(height)/2 = (2x + 10 + 10)(√(100 - x²))/2
    • A'(x) = (x + 10)(√(100 - x²)) - (2x² + 10x - 100)/√(100 - x²)
    • A'(x) = 0 gives x = 5
    • A''(x) = (-2x² - 10x + 100)/√(100 - x²)
    • A''(5) = -5 < 0, so x = 5 is a point of local maxima
    • The area of the trapezium is maximum at x = 5

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    This chapter explores the various applications of derivatives in different fields, building upon the concepts learned in Chapter 5. It covers the use of derivatives in understanding natural phenomena and solving problems.

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