2. Applications of Derivatives
  3. All Exercises, Answers, and Solutions

Exercises


Higher Order Derivatives


  1. Find \( f'(x), f''(x), \) and \( f'''(x) \) for the following functions.
    1. \( f(x) = (2 - 3x)^5 \)
    2. \( f(x) = x^{12} + 3x^4 \)
    3. \( f(x) = x^2 - \dfrac{1}{x} \)
    4. \( f(x) = \dfrac{4}{(x - 3)^2} \)
  2. Find \( \dfrac{dy}{dx} \) and \( \dfrac{d^2y}{dx^2} \) for the following curves:
    1. \( y = \sqrt[3]{x} - \dfrac{1}{\sqrt[3]{x^2}} \)
    2. \( y=(x^2 + 2)\sqrt{x} \)
    1. Given \( f'(x) = x^2, \) find \( f''(x) \).
    2. Given \( f''(x) = 3 - \dfrac{3}{x}, \) find \( f'''(x) \).
    3. Given \( f^{(4)}(x) = 2x + 1, \) find \( f^{(6)}(x) \).
  4. Determine the second derivative of the following functions.
    1. \( f(x) = 8x^{\frac{3}{2}} \)
    2. \( f(x) = \dfrac{x}{x - 1} \)
    3. \( f(x) = (x^3 + 2x)^4 \)
  5. Consider the function \( f(x) = g(x) h(x) \). Use the product rule to generate rules for finding \( f''(x) \) and \( f'''(x) \).
  6. Suppose \( f(x) = ax^2 + bx + c \) and \( f(1) = 8, f'(1) = 3, \) and \( f''(1) = -4 \). Determine \( a, b, \) and \( c \).
  7. Find \( \dfrac{d^{100}y}{dx^{100}} \) if \( y = x^{99} + 3x^{66} + 2x^{33} + 1 \).
  8. Develop a general formula for \( f^{(n)} (x) \) if
    1. \( f(x) = \dfrac{1}{x} \)
    2. \( f(x) = \dfrac{1}{3 - x} \)

Sketching the First and Second Derivative Functions


  1. f(x) is parabola with zeros x=0,4; f'(x) is line with slope 2, x-int 2; f''(x) is line y=2 The graphs of \( y = f(x), y = f'(x) \), and \( y = f''(x) \) are shown. Identify each curve and explain your choices.
     
  2. A:f increasing, curve gets less steep; B:f attains highest value in a small interval; C: f decreasing, curve gets less steep; D: f attains lowest value in a small interval; e: f increasing, curve gets more steep The graph of a function \(f(x)\) is shown and five values of \(x\) are indicated on the \(x\)-axis.
    1. At which of these values does \( f \) have a local maximum?
    2. At which of these values does \( f \) have a local minimum?
    3. Among these five values, where does \( f'(x) \) have the greatest value?
    4. Among these five values, where does \( f'(x) \) have the least value?
    5. At which of these values is \( f'(x) = 0 \)?
     
  3. Given the graph of \( y = f(x) \) below, fill in the table, indicating whether each of \( f(x), f'(x), f''(x) \) is negative, positive, or zero at each of the given points. A: decreasing, positive, concave up; B: decreasing, negative, concave up; C: local minimum, negative; D: negative, increasing, concave up; E: positive, increasing, concave down; F: local maximum, concave down; G: decreasing, positive, concave down
    Point \( f(x) \) \( f'(x) \) \( f''(x) \)
    \( A \)      
    \( B \)      
    \( C \)      
    \( D \)      
    \( E \)      
    \( F \)      
    \( G \)      
  4. Match the graphs of \( y = f(x), y = g(x) \), and \( y = h(x) \) with the graphs of \( y = f'(x), y = g'(x) \), and \( y = h'(x) \).
    \(y=f(x)\)Even degree, positive leading coefficient
    \(y=g(x)\)Odd degree, positive leading coefficient
     
    \(y=h(x)\)Even degree, positive leading coefficient
     
    a.Even degree, negative leading coefficient
    b.Odd degree, positive leading coefficient
     
    c.Even degree, positive leading coefficient
  5. Parabola with vertex at (0.5, 2.25), positive leading coefficient, zeros at x=-1,2 The graph of \( y = h'(x) \) for some cubic function \( h(x) \) is shown to the right. The tangent line to the curve \( y = h'(x) \) at \( x = -1 \) is drawn.
    1. At which values of \( x \), if any, does \( h(x) \) have a local maximum or minimum? Explain.
    2. On which intervals is \( h''(x) \) positive, negative, or zero? Explain.
    3. Sketch the graph of \( y = h''(x) \) on the same set of axes.
     
  6. Using the given graph of a polynomial \( f(x) \), sketch the graphs of \( f'(x) \) and \( f''(x) \) on the same set of axes.

    1. Parabola, vertex at (2, -2), zeros at x=0,4

    2. Cubic, double root at x=-1, zero at x=3.5, turning point at (2, -2.5), positive leading coefficient
  7. Using the given graph of \( g(x) \), sketch the graph of \( g'(x) \) on the same set of axes.
    Quartic, positive leading coefficient, turning points at (-2,0), (0,8), (2,0)
  8. Using the graph of \( f(x) = \left\lvert x^2 - 4 \right\rvert - 2 \), sketch the graph of \( y = f'(x) \) and \( f''(x) \) on the same set of axes. f(x)=-x^2 + 2 for -2 < x < 2, f(x)=x^2-6 elsewhere
  9. For each of the following sets of conditions, sketch a graph of a quadratic function \( f(x) \) which satisfies them.
      • \( f(-2) = f(2) = 0 \)
      • \( f'(x) \lt 0 \) for \( x \lt 0 \)
      • \( f'(0) = 0 \)
      • \( f'(x) \gt 0 \) for \( x \gt 0 \)
      • \( f''(x) \gt 0 \) for all \( x \in \mathbb{R} \)
      • \( f''(x) = -2 \) for all \( x \in \mathbb{R} \)
      • \( f'(1) = 0 \)
      • \( f(1) = 4 \)

Applications of Rates of Change: Geometry, Life Sciences, and Social Sciences


  1. Consider a triangle with base \( x \) and height \( 2x \). Find the instantaneous rate of change of the area of the triangle with respect to \( x \) when \( x = 5 \).
  2. Suppose the price of a commodity is given by the quadratic function \( P(t) = 3.78 + 0.25t + 0.2t^2 \). What is the instantaneous rate of change in the price when \( t = 5 \)?
  3. Suppose the cost \( C(q) \) (in dollars) of producing a quantity, \( q \), of a product is given by\[ C(q) = 500 + 3q + \dfrac{1}{10}q^2 \] The marginal cost, \( M(q) \), equals the instantaneous rate of change of the cost with respect to the quantity. Find the marginal cost when a quantity of \( 25 \) items is being produced.
  4. Let \( R(t) \) be the number of centimetres of water that has fallen since midnight, where \( t \) is measured in hours. Interpret the following.
    1. \( R(9) = 2.5 \)
    2. \( R'(9) = 0.2 \)
  5. If the cost, \( C \) (in cents), of purifying a litre of water to a purity of \( n \) percent is \( C(n) = \dfrac{100}{100 - n} \) for \( 50 \leq n \lt 100 \):
    1. Find the instantaneous rate of change of the cost with respect to the purity, \( n \).
    2. Find \( C'(92) \). Interpret your results.
    3. Find \( C'(97) \). Interpret your results.
  6. A company can produce widgets at a cost of \( $ 10 \) each, plus a fixed cost of \( $ 75 \). Therefore, the company's cost function is \( C(x) = 10x + 75 \), where \( x \) is the number of widgets produced.
    1. Find the average cost function.
    2. Find the marginal average cost function.
    3. Evaluate the marginal average cost function at \( x = 20 \). Interpret your result.
  7. The area of a rectangle is changing with respect to time. The width of the rectangle is given by \( w(t) = 3t - 1 \) and the height of the rectangle is given by \( h(t) = \sqrt{t} \), where \( t \) is measured in seconds and \( w\) and \(h \) are in centimetres. Find the rate of change of the area with respect to time when \( t = 4 \).
  8. The manager of an appliance manufacturing firm determines that when toasters are priced at \( p \) dollars apiece, the number sold each month can be modeled by\[ A(p) = \frac{300}{p} \] The manager estimates that \( t \) months from now, the unit price of the toasters will be \( p(t) = 0.04t^2 + 18.5 \) dollars. At what rate will the monthly demand for toasters, \( A(p) \), be changing two years from now? Will it be increasing or decreasing at this time?
  9. The radius of a right circular cylinder is given by \( \sqrt{t + 7} \) and its height is \( 2\sqrt{t} \), where \( t \) is measured in seconds and the dimensions are in metres. Find the rate of change of the volume when the height is \( 6~\text{m} \).

Applications of the Derivative to Motion


  1. A freight train leaves a train station and travels due north on a straight track.  After \( t \) hours, the train is \( s(t) = 18t^2 - 2t^3 \) kilometres north of the train station (for \( 0 \leq t \leq 9 \)).
    1. Find an expression for the velocity of the train at any time \( 0 \leq t \leq 9 \).
    2. Find the velocity of the train after \( 3 \) hours, and after \( 7 \) hours.
    3. Find the acceleration of the train after \( 1 \) hour.
  2. The position of a moving object at time \( t \) is given by \( s(t) = 2t^3 + 3t^2 - 12t, \ 0 \leq t \leq 10 \).
    1. Find the time interval(s) over which the object's velocity is positive.
    2. Find the time interval(s) over which the object's acceleration is positive.
  3. Each of the graphs in the figure below shows the position of a p article moving along the \( x \)-axis as a function of time, \( 0 \leq t \leq 5 \). During this time interval, which particle has
    1. a constant velocity
    2. the greatest initial velocity
    3. the greatest average velocity
    4. zero average velocity
    5. zero acceleration
    6. a negative acceleration for \( 0 \leq t \leq 5 \)

  1. Graph of s(t)=0.5*t-0.5, t between 0 and 5

  2. Graph of s(t)=(1/3)*sqrt(t), t between 0 and 5

  3. Graph of cubic with positive leading coefficient and zeros at t=0,2.5,5, t between 0 and 5

  4. Graph of exponential function passing with x-intercept at x=3.5, y-intercept -1
  1. A particle is moving along a straight line and its position, in metres, ahead of a fixed point on the line after \( t \) seconds, is given by the equation \( s = 18t - 12t^2 + 2t^3 \).
    1. Find the velocity and the acceleration of the particle after \( 3 \) seconds.
    2. Find the position(s) of the particle when its velocity is zero.
  2. A sailboat that is coasting with its sails down has velocity \( v(t) = \dfrac{100}{2t + 1} \) metres per second, at time \( t \).
    1. Give a formula for the acceleration of the sailboat as a function of time.
    2. When is the sailboat decelerating at \( 0.5~\text{m/s}^2 \)?
  3. A softball is thrown upwards. Its height, in metres, above the ground, \( t \) seconds after being released, is given by \( h(t) = 15t - 5t^2, t \geq 0 \).
    1. Find a formula for the vertical acceleration of the softball at time \( t \).
    2. What is the initial vertical velocity of the softball?
    3. What is the vertical velocity of the softball at the instant it hits the ground?
  4. A car is \( 80t + \dfrac{1}{10}t^5 \) kilometres past a service station at time \( t \) hours. Where is the car, relative to the service station, and how fast is it traveling, when its acceleration is \( 16~\text{km/h}^2 \)?
  5. The position function of a marble moving along a track is \( s(t) = (3 - 2t^2)t^{3/2} \), at time \( t \), in seconds.
    1. Find the marble's velocity and acceleration at time \( t \).
    2. When does the marble return to its starting position, \( s(0) \)?
    3. When is the marble at rest?
  6. A pumpkin is dropped from a dormitory window \( 15 \) metres above the ground. If there were no air resistance, it would fall \( s = 4.9t^2 \) metres in \( t \) seconds. What is the downward velocity and downward acceleration of the pumpkin, to one decimal place, when it is \( 5.2 \) metres from the ground?
  7. Most people expect elevator rides to be smooth, including when they start and stop.  This will be the case if the acceleration function is continuous.
    1. Show that the acceleration is continuous for all \( t \geq 0 \) for the position function \( s(t) = \begin{cases} 0 & \text{if } t = 0 \\ \dfrac{t^3}{t^2 + 1} & \text{if } t \gt 0 \end{cases} \).
    2. What happens to the velocity and acceleration for large values of \( t \)?