Learn – Calculus – Applications of Derivatives

Increasing and Decreasing functions

Problem set

In each of the following problems, give whether the function f(x) goes up or down at the given x value.

  1. At x=2, if f'(2) = 5
  2. At x=-5, if f'(-5) = 0.2
  3. At x=4, if f'(4) = -0.5
  4. At x=0, if f'(0) = -3
  5. At x=7, if f'(4) = -0.4
  6. At x=5, if f'(5) = 0

Problem set

In each of the following problems, give whether the given function goes up or down at the given x value.

  1. f(x) = x^3+x^4 at x=-2
  2. f(x) = x^4-x^6 at x=-0.1

Problem set

In each of the following problems, identify the intervals where the function is increasing and decreasing.

  1. 5x^2+4x+7
  2. 12x^3-4x
  3. x^3-x^2-5x+7

Minima and maxima

Problem set

For each of the following problems, graph the given function (approximately) and estimate where the extreme values of the function occur. Using calculus, find where the extreme values actually occur.

  1. x^3-4x
  2. (x-1)(x+1)(x+5)
  3. \frac{x}{x^2+2x+3}
  4. \sin x + \cos x in [0, 2\pi]
  5. 2\sin x - x in [0, 2\pi]

Problem set

For each of the following functions, identify the intervals where the function increases, the intervals where the function decreases, the local extreme values and the absolute extreme values.

  1. 16x^4-1
  2. 2x^3-4x^2+2x+7
  3. 2(x-1)^6-5
  4. -3(3x+2)^4+7
  5. 3x^3-5

Problem set

For each of the following functions, identify the intervals where the function increases, the intervals where the function decreases, the local extreme values and the absolute extreme values.

  1. \frac{9x+6}{3x^2+7}
  2. \frac{x-1}{x^2-x+16}
  3. \sqrt[3]{x}(x^2-28)
  4. \sqrt[3]{x^2}(x+10)
  5. x\sqrt{18-x^2}
  6. 2x^2\sqrt{10-x}

Problem set

For each of the following functions, identify the intervals where the function increases, the intervals where the function decreases, the local extreme values and the absolute extreme values.

  1. (4x^2-5)e^x
  2. (18-3x-2x^2)e^x
  3. e^x+e^{-x}
  4. e^x+3e^{-x} -2x
  5. 2e^x-6e^{-x} -13x
  6. 9^x-6\times 3^x
  7. \frac{1}{e^x+e^{-x}}

Problem set

For each of the following functions, identify the intervals where the function increases, the intervals where the function decreases, the local extreme values and the absolute extreme values.

  1. x\ln x
  2. x^2\ln 2x
  3. \frac{x^2}{4}(5-2\ln 2x)
  4. 5\ln x+8x-2x^2
  5. x^x, for x>0
  6. x^{1/x}, for x>0

Problem set

Find the local and absolute extreme values of the following functions.

  1. x^3 in [-1,1]
  2. -x^{1/3} in [-8, 8]
  3. x^{1/3}+\cos x in [-6, 6]
  4. 2x-x\ln x in [1, 10]
  5. \begin{cases} x + 1 & \mbox{ if }x \le 1 \\ (x-1)^2+2 & \mbox{ if }x > 1\end{cases}
  6. \frac{x^3}{e^x}

Problem set

Find the local and absolute extreme values of the following functions.

  1. |2x+1|
  2. |3x-5|-4
  3. |x^2-4|
  4. |16-x^2|-2
  5. |\sin x|
  6. |\sin x + \cos x|

Problem set

  1. If a = 3b-7, find the minimum value taken by ab.
  2. Find the two real numbers, that have a difference of 100 between them, making the smallest product.
  3. Find the dimensions of the rectangle with perimeter 100 units that has the biggest area.
  4. Find the dimensions of a right triangle with hypotenuse 8 units that has the biggest area.
  5. Find the biggest area possible for a right triangle that has a hypotenuse of 10 units.
  6. A right triangle has an area of 50 square units. What is the smallest possible length of the hypotenuse?
  7. Justify that of all the rectangles that you can form with a certain perimeter, a square has the largest area.
  8. Justify that of all the right-triangles that you can form with a certain hypotenuse, the isosceles right triangle has the largest area.
  9. A farmer bought a fence of length 400'\, . He wants to create a fenced rectangular area. What is the maximum area that he can form using the fence that he bought?
  10. A farmer wants to created a fenced rectangular area of 100 square feet. Determiner the dimensions of the rectangular area so that he needs to buy least length of the fence.

Concavity and graph sketching

Problem set

For each of the following functions, graph the function by determining the intercepts (if possible), horizontal asymptotes, vertical asymptotes, minimums, maximums, concavity and points of inflection.

  1. y=x^4-8x^3+5
  2. y=2x^3-5x^2+4x+11
  3. y=2x(x-4)^3
  4. y=x(x-5)^4

Problem set

For each of the following functions, graph the function by determining the intercepts (if possible), horizontal asymptotes, vertical asymptotes, minimums, maximums, concavity and points of inflection.

  1. y=\frac{4}{x^2-4}
  2. y=\frac{8}{x^2+4}
  3. y=\frac{x}{x^2+1}
  4. y=\frac{3x}{x^2+9}

Problem set

For each of the following functions, graph the function by determining the intercepts (if possible), horizontal asymptotes, vertical asymptotes, minimums, maximums, concavity and points of inflection.

  1. y=4x\sqrt{16-x^2}
  2. y=\sqrt{9-x^2}
  3. y=\frac{1}{\sqrt{x^2+1}}
  4. y=\frac{5x}{\sqrt{x^2+25}}

Problem set

For each of the following functions, graph the function by determining the intercepts (if possible), horizontal asymptotes, vertical asymptotes, minimums, maximums, concavity and points of inflection.

  1. y=\frac{x^3}{e^x}
  2. e^{\sin x} in [0,2\pi]
  3. \left(\ln x\right)^2

Problem set

For each of the following functions, graph the function by determining the intercepts (if possible), horizontal asymptotes, vertical asymptotes, minimums, maximums, concavity and points of inflection.

  1. y=|x^2-4|
  2. y=|2x^2-5x|

Problem set

In each of the following, the expression for f'(x) is given. For each of the following, determine the minimum/maximum, concavity and points of inflection for the corresponding f(x), and draw a possible graph of f(x).

  1. f'(x) = 3x^2-4x+1
  2. f'(x) = 6+5x-x^2
  3. f'(x) = 2x^3-16x^2
  4. f'(x) = 9x^2-3x^3

Problem set

In each of the following, the graph of f'(x) is shown. For each of the following, determine the minimum/maximum, concavity and points of inflection for the corresponding f(x), and draw a possible graph of f(x).
















What can you infer from derivatives?

Problem set

In the following, assume f is a twice differentiable function – that is, f''(x) exists everywhere in the domain of f. Identify with justification whether each of the following statements is true or false.

  1. If f''(2) > 0, then f is increasing at 2.
  2. If f''(x) < 0 for all x, then it is possible for f to have a local minimum.
  3. If f'(-5) = 0 and f'(x) > 0 for x > -5, then f has a minimum at -5.
  4. If f'(x) = (x-1)(9-x), then f has a maximum at x = 9.
  5. If f''(-1) < 0, then f is decreasing at -1.

Problem set

In the following, assume f is a twice differentiable function – that is, f''(x) exists everywhere in the domain of f. Identify with justification whether each of the following statements is true or false.

  1. If f'(x) has a minimum at 10.1, then f has a point of inflection at 10.1
  2. If f'(3.2) = 0, then f has an extremum at 3.2.
  3. If f''(-10) = 0, then f has a point of inflection at -10.
  4. If f'(-2.9) = 0, and f''(-2.9) < 0, then f has a point of inflection at -2.9
  5. If f'(4) = 0, then f has a point of inflection at 4.

Problem set

In the following, assume f is a twice differentiable function – that is, f''(x) exists everywhere in the domain of f. Identify with justification whether each of the following statements is true or false.

  1. If f'(-5) = 0, f'(x) > 0 for x > -5 and f'(x) < 0 for x < -5, then f has a minimum at -5.
  2. If f'(x) = (x-1)(9-x)^2, then f has a maximum at x = 9.
  3. If f''(x) < 0 for all x, then it is possible for f to have a local maximum.
  4. If f''(4) = 0, then f has an extremum at 4.
  5. If f'(x) has a maximum at 10.1, then f has a point of inflection at 10.1

Problem set

In the following, assume f is a twice differentiable function – that is, f''(x) exists everywhere in the domain of f. Identify with justification whether each of the following statements is true or false.

  1. If f'(x) has a minimum at 10.1, then f concaves up to the immediate right of 10.1
  2. If f'(-3.1) = 0 and f''(-3.1) = 0, then f has an extremum at -3.1.
  3. If f''(0) = 0, then f has a point of inflection at 0.
  4. If f has a point of inflection at 3, then f''(3) = 0.
  5. If f has a point of inflection at 3, then f has an extremum at 3.

Related rates

Problem set

  1. A spherical ballon is being filled with air at the rate of 10 cubic feet/min. When the radius of the balloon is 5 feet,

    1. at what rate is the radius of the balloon increasing?

    2. at what rate is the surface area of the balloon increasing?

  2. Main street and Spring street are two streets that cross each other perpendicularly and that stretch long. Sachi and Salim start biking at the intersection of the two streets, with Sachi taking Main street and Salim taking Spring street. If Sachi bikes at 10 miles per hour and Salim bikes at 8 miles per hour, at what rate is the distance between them changing 1 hour after start?

  3. The view-point of a rocket launch is 1 km from the launch pad. If the rocket blasts off vertically at 2 km/sec, at what rate is the distance between the rocket and the view-point changing when the rocket is 4 km above the ground?

  4. A conical water tank (vertex down) is of height 10 feet and radius 4 feet. If water leaks from the tank at a constant rate of 2 cubic feet/min, how fast is the level of water decreasing when the height of water is 8 feet?

  5. Tabu is 6 feet tall and is walking at a speed of 4 feet/sec towards a street lamp that is 20 feet above the ground.

    1. At what rate is the tip of her shadow moving?

    2. At what rate is the length of the shadow changing when she is 8 feet from the base of the lamp?

Real-world applications

Problem set

In each of the following problems, the position of a particle moving along the X-axis is given as a function of time. For each of the problems, describe the motion of the particle by doing the following:

  • identify the time instant when the particle changed direction
  • the maximum distance the particle would be from the origin
  • the time intervals when the particle is moving away from the origin and moving towards the origin
  • the time-intervals when the particle is accelerating or decelerating
  1. x(t) = t^3 +2t^2 -4t-10
  2. x(t) = 4t^3 - 4t^2 + t - 2
  3. x(t) = 4t^3 - 5t^2 + 2t - 5
  4. x(t) = 5t^4 - t^3 - t^2 + 2







Problem set

  1. Use tables giving the behavior of a function such as horizontal tangent, concavity, rising, falling etc to draw the graph of a function.
  2. Use tables giving values of derivative and the second derivative to sketch the graph of a function.