Introduction to Derivatives: Get Stronger

Defining the Derivative

For the following exercises (1-5), use the definition of a derivative to find the slope of the secant line between the values [latex]x_1[/latex] and [latex]x_2[/latex] for each function [latex]y=f(x)[/latex].

  1. [latex]f(x)=4x+7; \,\,\, x_1=2, \,\,\, x_2=5[/latex]
  2. [latex]f(x)=x^2+2x+1; \,\,\, x_1=3, \,\,\, x_2=3.5[/latex]
  3. [latex]f(x)=\dfrac{4}{3x-1}; \,\,\, x_1=1, \,\,\, x_2=3[/latex]
  4. [latex]f(x)=\sqrt{x}; \,\,\, x_1=1, \,\,\, x_2=16[/latex]
  5. [latex]f(x)=x^{\frac{1}{3}}+1; \,\,\, x_1=0, \,\,\, x_2=8[/latex]

For the following functions (6-10),

  1. Use [latex]m_{\tan}=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}[/latex] to find the slope of the tangent line [latex]m_{\tan}=f^{\prime}(a)[/latex], and
  2. find the equation of the tangent line to [latex]f[/latex] at [latex]x=a[/latex].
  1. [latex]f(x)=3-4x, \,\,\, a=2[/latex]
  2. [latex]f(x)=x^2+x, \,\,\, a=1[/latex]
  3. [latex]f(x)=\dfrac{7}{x}, \,\,\, a=3[/latex]
  4. [latex]f(x)=2-3x^2, \,\,\, a=-2[/latex]
  5. [latex]f(x)=\dfrac{2}{x+3}, \,\,\, a=-4[/latex]

For the following functions [latex]y=f(x)[/latex] (11-15), find [latex]f^{\prime}(a)[/latex] using [latex]f^{\prime}(a)=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}[/latex].

  1. [latex]f(x)=5x+4, \,\,\, a=-1[/latex]
  2. [latex]f(x)=x^2+9x, \,\,\, a=2[/latex]
  3. [latex]f(x)=\sqrt{x}, \,\,\, a=4[/latex]
  4. [latex]f(x)=\dfrac{1}{x}, \,\,\, a=2[/latex]
  5. [latex]f(x)=\dfrac{1}{x^3}, \,\,\, a=1[/latex]

For the following exercises (16-17), given the function [latex]y=f(x)[/latex],

  1. find the slope of the secant line [latex]PQ[/latex] for each point [latex]Q(x,f(x))[/latex] with [latex]x[/latex] value given in the table.
  2. Use the answers from a. to estimate the value of the slope of the tangent line at [latex]P[/latex].
  3. Use the answer from b. to find the equation of the tangent line to [latex]f[/latex] at point [latex]P[/latex].
  1. [latex]f(x)=x^2+3x+4, \,\,\, P(1,8)[/latex] (Round to 6 decimal places.)
    [latex]x[/latex] Slope [latex]m_{PQ}[/latex] [latex]x[/latex] Slope [latex]m_{PQ}[/latex]
    [latex]1.1[/latex] (i) [latex]0.9[/latex] (vii)
    [latex]1.01[/latex] (ii) [latex]0.99[/latex] (viii)
    [latex]1.001[/latex] (iii) [latex]0.999[/latex] (ix)
    [latex]1.0001[/latex] (iv) [latex]0.9999[/latex] (x)
    [latex]1.00001[/latex] (v) [latex]0.99999[/latex] (xi)
    [latex]1.000001[/latex] (vi) [latex]0.999999[/latex] (xii)
  2. [latex]f(x)=10e^{0.5x}, \,\,\, P(0,10)[/latex] (Round to 4 decimal places.)
    [latex]x[/latex] Slope [latex]m_{PQ}[/latex]
    [latex]-0.1[/latex] (i)
    [latex]-0.01[/latex] (ii)
    [latex]-0.001[/latex] (iii)
    [latex]-0.0001[/latex] (iv)
    [latex]-0.00001[/latex] (v)
    [latex]−0.000001[/latex] (vi)

For the following position functions [latex]y=s(t)[/latex] (18-19), an object is moving along a straight line, where [latex]t[/latex] is in seconds and [latex]s[/latex] is in meters. Find

  1. the simplified expression for the average velocity from [latex]t=2[/latex] to [latex]t=2+h[/latex];
  2. the average velocity between [latex]t=2[/latex] and [latex]t=2+h[/latex], where (i) [latex]h=0.1[/latex], (ii) [latex]h=0.01[/latex], (iii) [latex]h=0.001[/latex], and (iv) [latex]h=0.0001[/latex]; and
  3. use the answer from a. to estimate the instantaneous velocity at [latex]t=2[/latex] seconds.
  1. [latex]s(t)=\frac{1}{3}t+5[/latex]
  2. [latex]s(t)=2t^3+3[/latex]

For the following exercises (20-21), use the limit definition of derivative to show that the derivative does not exist at [latex]x=a[/latex] for each of the given functions.

  1. [latex]f(x)=x^{\frac{1}{3}}, \, x=0[/latex]
  2. [latex]f(x)=\begin{cases} 1 & \text{ if } \, x<1 \\ x & \text{ if } \, x \ge 1 \end{cases}, \, x=1[/latex]

For the following exercises (22-26), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. The position in feet of a race car along a straight track after [latex]t[/latex] seconds is modeled by the function [latex]s(t)=8t^2-\frac{1}{16}t^3[/latex].
    1. Find the average velocity of the vehicle over the following time intervals to four decimal places:
      1. [latex][4, 4.1][/latex]
      2. [latex][4, 4.01][/latex]
      3. [latex][4, 4.001][/latex]
      4. [latex][4, 4.0001][/latex]
    2. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at [latex]t=4[/latex] seconds.
  2. Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by [latex]s=f(t)[/latex] and [latex]s=g(t)[/latex], where [latex]s[/latex] is measured in feet and [latex]t[/latex] is measured in seconds.

    Two functions s = g(t) and s = f(t) are graphed. The first function s = g(t) starts at (0, 0) and arcs upward through roughly (2, 1) to (4, 4). The second function s = f(t) is a straight line passing through (0, 0) and (4, 4).

    1. Which vehicle has traveled farther at [latex]t=2[/latex] seconds?
    2. What is the approximate velocity of each vehicle at [latex]t=3[/latex] seconds?
    3. Which vehicle is traveling faster at [latex]t=4[/latex] seconds?
    4. What is true about the positions of the vehicles at [latex]t=4[/latex] seconds?
  3. For the function [latex]f(x)=x^3-2x^2-11x+12[/latex], do the following.
    1. Use a graphing calculator to graph [latex]f[/latex] in an appropriate viewing window.
    2. Use the ZOOM feature on the calculator to approximate the two values of [latex]x=a[/latex] for which [latex]m_{\tan}=f^{\prime}(a)=0[/latex].
  4. Suppose that [latex]N(x)[/latex] computes the number of gallons of gas used by a vehicle traveling [latex]x[/latex] miles. Suppose the vehicle gets 30 mpg.
    1. Find a mathematical expression for [latex]N(x)[/latex].
    2. What is [latex]N(100)[/latex]? Explain the physical meaning.
    3. What is [latex]N^{\prime}(100)[/latex]? Explain the physical meaning.
  5. For the function [latex]f(x)=\dfrac{x^2}{x^2+1}[/latex], do the following.
    1. Use a graphing calculator to graph [latex]f[/latex] in an appropriate viewing window.
    2. Use the [latex]\text{nDeriv}[/latex] function on a graphing calculator to find [latex]f^{\prime}(-4), \, f^{\prime}(-2), \, f^{\prime}(2)[/latex], and [latex]f^{\prime}(4)[/latex].

The Derivative as a Function

For the following exercises (1-5), use the definition of a derivative to find [latex]f^{\prime}(x)[/latex].

  1. [latex]f(x)=2-3x[/latex]
  2. [latex]f(x)=4x^2[/latex]
  3. [latex]f(x)=\sqrt{2x}[/latex]
  4. [latex]f(x)=\dfrac{9}{x}[/latex]
  5. [latex]f(x)=\dfrac{1}{\sqrt{x}}[/latex]

For the following exercises (6-7), use the graph of [latex]y=f(x)[/latex] to sketch the graph of its derivative [latex]f^{\prime}(x)[/latex].

  1. The function f(x) starts at (−1.5, 20) and decreases to pass through (0, 10), where it appears to have a derivative of 0. Then it further decreases, passing through (1.7, 0) and achieving a minimum at (3, −17), at which point it increases rapidly through (3.8, 0) to (4, 20).
  2. The function f(x) starts at (−3, −1) and increases to pass through (−1.5, 0) and achieve a local minimum at (1, 0). Then, it decreases and passes through (1.5, 0) and continues decreasing to (3, −1).

For the following exercises (8-10), the given limit represents the derivative of a function [latex]y=f(x)[/latex] at [latex]x=a[/latex]. Find [latex]f(x)[/latex] and [latex]a[/latex].

  1. [latex]\underset{h\to 0}{\lim}\dfrac{[3(2+h)^2+2]-14}{h}[/latex]
  2. [latex]\underset{h\to 0}{\lim}\dfrac{(2+h)^4-16}{h}[/latex]
  3. [latex]\underset{h\to 0}{\lim}\dfrac{e^h-1}{h}[/latex]

For the following functions (11-12),

  1. sketch the graph and
  2. use the definition of a derivative to show that the function is not differentiable at [latex]x=1[/latex].
  1. [latex]f(x)=\begin{cases} 3 & \text{ if } \, x<1 \\ 3x & \text{ if } \, x \ge 1 \end{cases}[/latex]
  2. [latex]f(x)=\begin{cases} 2x & \text{ if } x \le 1 \\ \dfrac{2}{x} & \text{ if } \, x>1 \end{cases}[/latex]

For the following functions (13-14), use [latex]f''(x)=\underset{h\to 0}{\lim}\dfrac{f^{\prime}(x+h)-f^{\prime}(x)}{h}[/latex] to find [latex]f''(x)[/latex].

  1. [latex]f(x)=2-3x[/latex]
  2. [latex]f(x)=x+\dfrac{1}{x}[/latex]

For the following exercises (15-17), use a calculator to graph [latex]f(x)[/latex]. Determine the function [latex]f^{\prime}(x)[/latex], then use a calculator to graph [latex]f^{\prime}(x)[/latex].

  1. [latex]f(x)=3x^2+2x+4[/latex]
  2. [latex]f(x)=\dfrac{1}{\sqrt{2x}}[/latex]
  3. [latex]f(x)=x^3+1[/latex]

For the following exercises (18-20), describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

  1. [latex]\dfrac{f(x+h)-f(x)}{h}[/latex]
  2. [latex]f^{\prime}(x)=\underset{h\to 0}{\lim}\dfrac{f(x+h)-f(x)}{h}[/latex]
  1. [latex]C(x)[/latex] denotes the total amount of money (in thousands of dollars) spent on concessions by [latex]x[/latex] customers at an amusement park.
  2. [latex]g(x)[/latex] denotes the grade (in percentage points) received on a test, given [latex]x[/latex] hours of studying.
  3. [latex]p(x)[/latex] denotes atmospheric pressure at an altitude of [latex]x[/latex] feet.

For the following exercises (21-22), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. Suppose temperature [latex]T[/latex] in degrees Fahrenheit at a height [latex]x[/latex] in feet above the ground is given by [latex]y=T(x)[/latex].
    1. Give a physical interpretation, with units, of [latex]T^{\prime}(x)[/latex].
    2. If we know that [latex]{T}^{\prime }(1000)=-0.1,[/latex] explain the physical meaning.
  2. The graph in the following figure models the number of people [latex]N(t)[/latex] who have come down with the flu [latex]t[/latex] weeks after its initial outbreak in a town with a population of 50,000 citizens.
    1. Describe what [latex]N^{\prime}(t)[/latex] represents and how it behaves as [latex]t[/latex] increases.
    2. What does the derivative tell us about how this town is affected by the flu outbreak?

    The function starts at (0, 3000) and increases quickly to an asymptote at y = 50000.

For the following exercises (23-28), use the following table, which shows the height [latex]h[/latex] of the Saturn V rocket for the Apollo 11 mission [latex]t[/latex] seconds after launch.

Time (seconds) Height (meters)
[latex]0[/latex] [latex]0[/latex]
[latex]1[/latex] [latex]2[/latex]
[latex]2[/latex] [latex]4[/latex]
[latex]3[/latex] [latex]13[/latex]
[latex]4[/latex] [latex]25[/latex]
[latex]5[/latex] [latex]32[/latex]
  1. What is the physical meaning of [latex]h^{\prime}(t)[/latex]? What are the units?
  2. Construct a table of values for [latex]h^{\prime}(t)[/latex] and graph both [latex]h(t)[/latex] and [latex]h^{\prime}(t)[/latex] on the same graph. (Hint: for interior points, estimate both the left limit and right limit and average them. An interior point of an interval [latex]I[/latex] is an element of [latex]I[/latex] which is not an endpoint of [latex]I[/latex].)
  3. The best linear fit to the data is given by [latex]H(t)=7.229t-4.905[/latex], where [latex]H[/latex] is the height of the rocket (in meters) and [latex]t[/latex] is the time elapsed since takeoff. From this equation, determine [latex]H^{\prime}(t)[/latex]. Graph [latex]H(t)[/latex] with the given data and, on a separate coordinate plane, graph [latex]H^{\prime}(t)[/latex].
  4. The best quadratic fit to the data is given by [latex]G(t)=1.429t^2+0.0857t-0.1429[/latex], where [latex]G[/latex] is the height of the rocket (in meters) and [latex]t[/latex] is the time elapsed since takeoff. From this equation, determine [latex]G^{\prime}(t)[/latex]. Graph [latex]G(t)[/latex] with the given data and, on a separate coordinate plane, graph [latex]G^{\prime}(t)[/latex].
  5. The best cubic fit to the data is given by [latex]F(t)=0.2037t^3+2.956t^2-2.705t+0.4683[/latex], where [latex]F[/latex] is the height of the rocket (in m) and [latex]t[/latex] is the time elapsed since take off. From this equation, determine [latex]F^{\prime}(t)[/latex]. Graph [latex]F(t)[/latex] with the given data and, on a separate coordinate plane, graph [latex]F^{\prime}(t)[/latex]. Does the linear, quadratic, or cubic function fit the data best?
  6. Using the best linear, quadratic, and cubic fits to the data, determine what [latex]H''(t), \, G''(t)[/latex], and [latex]F''(t)[/latex] are. What are the physical meanings of [latex]H''(t), \, G''(t)[/latex], and [latex]F''(t)[/latex], and what are their units?

Differentiation Rules

For the following exercises (1-6), find [latex]f^{\prime}(x)[/latex] for each function.

  1. [latex]f(x)=5x^3-x+1[/latex]
  2. [latex]f(x)=8x^4+9x^2-1[/latex]
  3. [latex]f(x)=3x\left(18x^4+\dfrac{13}{x+1}\right)[/latex]
  4. [latex]f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)[/latex]
  5. [latex]f(x)=\dfrac{4x^3-2x+1}{x^2}[/latex]
  6. [latex]f(x)=\dfrac{x+9}{x^2-7x+1}[/latex]

For the following exercises (7-8), find the equation of the tangent line [latex]T(x)[/latex] to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.

  1. [latex]y=2\sqrt{x}+1[/latex]  at  [latex](4,5)[/latex]
  2. [latex]y=\dfrac{2}{x}-\dfrac{3}{x^2}[/latex]  at  [latex](1,-1)[/latex]

For the following exercises (9-10), assume that [latex]f(x)[/latex] and [latex]g(x)[/latex] are both differentiable functions for all [latex]x[/latex]. Find the derivative of each of the functions [latex]h(x)[/latex].

  1. [latex]h(x)=x^3f(x)[/latex]
  2. [latex]h(x)=\dfrac{3f(x)}{g(x)+2}[/latex]

For the following exercises (11-12), assume that [latex]f(x)[/latex] and [latex]g(x)[/latex] are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

[latex]x[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex]
[latex]f(x)[/latex] [latex]3[/latex] [latex]5[/latex] [latex]-2[/latex] [latex]0[/latex]
[latex]g(x)[/latex] [latex]2[/latex] [latex]3[/latex] [latex]-4[/latex] [latex]6[/latex]
[latex]f^{\prime}(x)[/latex] [latex]-1[/latex] [latex]7[/latex] [latex]8[/latex] [latex]-3[/latex]
[latex]g^{\prime}(x)[/latex] [latex]4[/latex] [latex]1[/latex] [latex]2[/latex] [latex]9[/latex]
  1. Find [latex]h^{\prime}(2)[/latex] if [latex]h(x)=\dfrac{f(x)}{g(x)}[/latex].
  2. Find [latex]h^{\prime}(4)[/latex] if [latex]h(x)=\dfrac{1}{x}+\dfrac{g(x)}{f(x)}[/latex].

For the following exercises (13-14),

  1. Evaluate [latex]f^{\prime}(a)[/latex], and
  2. Graph the function [latex]f(x)[/latex] and the tangent line at [latex]x=a[/latex].
  1. [latex]f(x)=2x^3+3x-x^2, \,\,\, a=2[/latex]
  2. [latex]f(x)=x^2-x^{12}+3x+2, \,\,\, a=0[/latex]

For the following exercises (15-21), solve each problem.

  1. Find the equation of the tangent line to the graph of [latex]f(x)=2x^3+4x^2-5x-3[/latex] at [latex]x=-1[/latex].
  2. Find the equation of the tangent line to the graph of [latex]f(x)=(3x-x^2)(3-x-x^2)[/latex] at [latex]x=1[/latex].
  3. Find the equation of the line passing through the point [latex]P(3,3)[/latex] and tangent to the graph of [latex]f(x)=\dfrac{6}{x-1}[/latex].
  4. Find a quadratic polynomial such that [latex]f(1)=5, \, f^{\prime}(1)=3[/latex], and [latex]f''(1)=-6[/latex].
  5. A herring swimming along a straight line has traveled [latex]s(t)=\dfrac{t^2}{t^2+2}[/latex] feet in [latex]t[/latex] seconds.

    Determine the velocity of the herring when it has traveled [latex]3[/latex] seconds.

  6. The concentration of antibiotic in the bloodstream [latex]t[/latex] hours after being injected is given by the function [latex]C(t)=\dfrac{2t^2+t}{t^3+50}[/latex], where [latex]C[/latex] is measured in milligrams per liter of blood.
    1. Find the rate of change of [latex]C(t)[/latex].
    2. Determine the rate of change for [latex]t=8, \, 12, \, 24[/latex], and [latex]36[/latex].
    3. Briefly describe what seems to be occurring as the number of hours increases.
  7. According to Newton’s law of universal gravitation, the force [latex]F[/latex] between two bodies of constant mass [latex]m_1[/latex] and [latex]m_2[/latex] is given by the formula [latex]F=\dfrac{G m_1 m_2}{d^2}[/latex], where [latex]G[/latex] is the gravitational constant and [latex]d[/latex] is the distance between the bodies.
    1. Suppose that [latex]G, \, m_1[/latex], and [latex]m_2[/latex] are constants. Find the rate of change of force [latex]F[/latex] with respect to distance [latex]d[/latex].
    2. Find the rate of change of force [latex]F[/latex] with gravitational constant [latex]G=6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2[/latex], on two bodies [latex]10[/latex] meters apart, each with a mass of [latex]1000[/latex] kilograms.

Derivatives as Rates of Change

  1. The function below represents the position of a particle traveling along a horizontal line.
    1. Find the velocity and acceleration functions.
    2. Determine the time intervals when the object is slowing down or speeding up.

    [latex]s(t)=2t^3-15t^2+36t-10[/latex]

  2. A rocket is fired vertically upward from the ground. The distance [latex]s[/latex] in feet that the rocket travels from the ground after [latex]t[/latex] seconds is given by [latex]s(t)=-16t^2+560t[/latex].
    1. Find the velocity of the rocket [latex]3[/latex] seconds after being fired.
    2. Find the acceleration of the rocket [latex]3[/latex] seconds after being fired.
  3. The position function [latex]s(t)=t^2-3t-4[/latex] represents the position of the back of a car backing out of a driveway and then driving in a straight line, where [latex]s[/latex] is in feet and [latex]t[/latex] is in seconds. In this case, [latex]s(t)=0[/latex] represents the time at which the back of the car is at the garage door, so [latex]s(0)=-4[/latex] is the starting position of the car, [latex]4[/latex] feet inside the garage.
    1. Determine the velocity of the car when [latex]s(t)=0[/latex].
    2. Determine the velocity of the car when [latex]s(t)=14[/latex].
  4. A potato is launched vertically upward with an initial velocity of [latex]100 ft/s[/latex] from a potato gun at the top of an [latex]85[/latex]-foot-tall building. The distance in feet that the potato travels from the ground after [latex]t[/latex] seconds is given by [latex]s(t)=-16t^2+100t+85[/latex].
    1. Find the velocity of the potato after [latex]0.5[/latex] sec and [latex]5.75[latex] sec.
    2. Find the speed of the potato at [latex]0.5[/latex] sec and [latex]5.75[latex] sec.
    3. Determine when the potato reaches its maximum height.
    4. Find the acceleration of the potato at [latex]0.5[/latex] s and [latex]1.5[/latex] s.
    5. Determine how long the potato is in the air.
    6. Determine the velocity of the potato upon hitting the ground.
  5. The following graph shows the position [latex]y=s(t)[/latex] of an object moving along a straight line. On the Cartesian coordinate plane, a function is graphed that is part of a parabola from the origin to (2, 2) with maximum at (1.5, 2.25). Then the function is constant until (5, 2), at which points becomes a parabola again, decreasing to a minimum at (6, 1) and then increasing to (7, 2).
    1. Use the graph of the position function to determine the time intervals when the velocity is positive, negative, or zero.
    2. Sketch the graph of the velocity function.
    3. Use the graph of the velocity function to determine the time intervals when the acceleration is positive, negative, or zero.
    4. Determine the time intervals when the object is speeding up or slowing down.
  6. The price [latex]p[/latex] (in dollars) and the demand [latex]x[/latex] for a certain digital clock radio is given by the price-demand function [latex]p=10-0.001x[/latex].
    1. Find the revenue function [latex]R(x)[/latex].
    2. Find the marginal revenue function.
    3. Find the marginal revenue at [latex]x=2000[/latex] and [latex]x=5000[/latex].
  7. In general, the profit function is the difference between the revenue and cost functions: [latex]P(x)=R(x)-C(x)[/latex].

    Suppose the price-demand and cost functions for the production of cordless drills is given respectively by [latex]p=143-0.03x[/latex] and [latex]C(x)=75,000+65x[/latex], where [latex]x[/latex] is the number of cordless drills that are sold at a price of [latex]p[/latex] dollars per drill and [latex]C(x)[/latex] is the cost of producing [latex]x[/latex] cordless drills.

    1. Find the marginal cost function.
    2. Find the revenue and marginal revenue functions.
    3. Find [latex]R^{\prime}(1000)[/latex] and [latex]R^{\prime}(4000)[/latex]. Interpret the results.
    4. Find the profit and marginal profit functions.
    5. Find [latex]P^{\prime}(1000)[/latex] and [latex]P^{\prime}(4000)[/latex]. Interpret the results.
  8. A culture of bacteria grows in number according to the function [latex]N(t)=3000\left(1+\dfrac{4t}{t^2+100}\right)[/latex], where [latex]t[/latex] is measured in hours.
    1. Find the rate of change of the number of bacteria.
    2. Find [latex]N^{\prime}(0), \, N^{\prime}(10), \, N^{\prime}(20)[/latex], and [latex]N^{\prime}(30)[/latex].
    3. Interpret the results in (b).
    4. Find [latex]N''(0), \, N''(10), \, N''(20)[/latex], and [latex]N''(30)[/latex]. Interpret what the answers imply about the bacteria population growth.