Analytical Applications of Derivatives: Get Stronger

Related Rates

For the following exercises (1-11), find the quantities for the given equation.

  1. Find [latex]\frac{dy}{dt}[/latex] at [latex]x=1[/latex] and [latex]y=x^2+3[/latex] if [latex]\frac{dx}{dt}=4[/latex].
  2. Find [latex]\frac{dz}{dt}[/latex] at [latex](x,y)=(1,3)[/latex] and [latex]z^2=x^2+y^2[/latex] if [latex]\frac{dx}{dt}=4[/latex] and [latex]\frac{dy}{dt}=3[/latex].
  3. A [latex]10[/latex] ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of [latex]2[/latex] ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is [latex]5[/latex] ft from the wall?

    A right triangle is formed by a ladder leaning up against a brick wall. The ladder forms the hypotenuse and is 10 ft long.

  4. Two airplanes are flying in the air at the same height: airplane [latex]A[/latex] is flying east at [latex]250[/latex] mi/h and airplane [latex]B[/latex] is flying north at [latex]300[/latex] mi/h.  If they are both heading to the same airport, located [latex]30[/latex] miles east of airplane [latex]A[/latex] and [latex]40[/latex] miles north of airplane [latex]B[/latex], at what rate is the distance between the airplanes changing?

    A right triangle is formed by two airplanes A and B moving perpendicularly to each other. The hypotenuse is the distance between planes A and B. The other sides are extensions of each plane’s path until they meet.

  5. Two buses are driving along parallel freeways that are [latex]5[/latex] mi apart, one heading east and the other heading west. Assuming that each bus drives a constant [latex]55[/latex] mph, find the rate at which the distance between the buses is changing when they are [latex]13[/latex] mi apart, heading toward each other.
  6. A [latex]6[/latex]-ft-tall person walks away from a [latex]10[/latex] ft lamppost at a constant rate of [latex]3[/latex] ft/sec. What is the rate that the tip of the shadow moves away from the pole when the person is [latex]10[/latex] ft away from the pole?

    A lamppost is shown that is 10 ft high. To its right, there is a person who is 6 ft tall. There is a line from the top of the lamppost that touches the top of the person’s head and then continues to the ground. The length from the end of this line to where the lamppost touches the ground is 10 + x. The distance from the lamppost to the person on the ground is 10, and the distance from the person to the end of the line is x.

  7. Using the previous problem, what is the rate at which the tip of the shadow moves away from the person when the person is [latex]10[/latex] ft from the pole?
  8. A [latex]5[/latex]-ft-tall person walks toward a wall at a rate of [latex]2[/latex] ft/sec. A spotlight is located on the ground [latex]40[/latex] ft from the wall. How fast does the height of the person’s shadow on the wall change when the person is [latex]10[/latex] ft from the wall?
  9. Using the previous problem, what is the rate at which the shadow changes when the person is [latex]10[/latex] ft from the wall, if the person is walking away from the wall at a rate of [latex]2[/latex] ft/sec?
  10. A helicopter starting on the ground is rising directly into the air at a rate of [latex]25[/latex] ft/sec. You are running on the ground starting directly under the helicopter at a rate of [latex]10[/latex] ft/sec. Find the rate of change of the distance between the helicopter and yourself after [latex]5[/latex] sec.
  11. Using the previous problem, what is the rate at which the distance between you and the helicopter is changing when the helicopter has risen to a height of [latex]60[/latex] ft in the air, assuming that, initially, it was [latex]30[/latex] ft above you?

For the following exercises (12-15), draw and label diagrams to help solve the related-rates problems.

  1. The volume of a cube decreases at a rate of [latex]10[/latex] m3/s. Find the rate at which the side of the cube changes when the side of the cube is [latex]2[/latex] m.
  2. The radius of a sphere decreases at a rate of [latex]3[/latex] m/sec. Find the rate at which the surface area decreases when the radius is [latex]10[/latex] m.
  3. The radius of a sphere is increasing at a rate of [latex]9[/latex] cm/sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.
  4. A triangle has two constant sides of length [latex]3[/latex] ft and [latex]5[/latex] ft. The angle between these two sides is increasing at a rate of [latex]0.1[/latex] rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is [latex]\pi /6[/latex].

For the following exercises (16-19), consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft.

  1. How fast does the depth of the water change when the water is [latex]10[/latex] ft high if the cone leaks water at a rate of [latex]10[/latex] [latex]\text{ft}^3[/latex]/min?
  2. If the water level is decreasing at a rate of [latex]3[/latex] in/min when the depth of the water is [latex]8[/latex] ft, determine the rate at which water is leaking out of the cone.
  3. A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of [latex]2[/latex] m and a radius of [latex]2[/latex] m. Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is [latex]10[/latex] cm/min when the height is [latex]1[/latex] m.
  4. A tank is shaped like an upside-down square pyramid, with base of [latex]4[/latex] m by [latex]4[/latex] m and a height of [latex]12[/latex] m (see the following figure). How fast does the height increase when the water is [latex]2[/latex] m deep if water is being pumped in at a rate of [latex]\frac{2}{3} \, \text{m}^3[/latex]/sec?

    An upside-down square pyramid is shown with square side lengths 4 and height 12. There is an unspecified amount of water inside the shape.

For the following exercises (20-22), draw the situations and solve the related-rate problems.

  1. You are stationary on the ground and are watching a bird fly horizontally at a rate of [latex]10[/latex] m/sec. The bird is located [latex]40[/latex] m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is [latex]9[/latex] m?
  2. A lighthouse, [latex]L[/latex], is on an island [latex]4[/latex] mi away from the closest point, [latex]P[/latex], on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of [latex]10[/latex] revolutions/min, how fast does the beam of light move across the beach [latex]2[/latex] mi away from the closest point on the beach?

    A right triangle is formed by a lighthouse L, a point P on the shore that is perpendicular to the line from the lighthouse to the shore, and a point 2 miles to the right of the point P. The distance from P to L is 4 miles.

  3. You are walking to a bus stop at a right-angle corner. You move north at a rate of [latex]2[/latex] m/sec and are [latex]20[/latex] m south of the intersection. The bus travels west at a rate of [latex]10[/latex] m/sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are [latex]20[/latex] m south of the intersection and the bus is [latex]10[/latex] m west of the intersection?

For the following exercises (23-24), refer to the figure of baseball diamond, which has sides of [latex]90[/latex] ft.

A baseball field is shown, with the bases labeled Home, 1st, 2nd, and 3rd making a square with side lengths 90 ft.

  1. A batter hits a ball toward second base at [latex]80[/latex] ft/sec and runs toward first base at a rate of [latex]30[/latex] ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)
  2. Runners start at first and second base. When the baseball is hit, the runner at first base runs at a speed of [latex]18[/latex] ft/sec toward second base and the runner at second base runs at a speed of [latex]20[/latex] ft/sec toward third base. How fast is the distance between runners changing [latex]1[/latex] sec after the ball is hit?

Linear Approximations and Differentials

  1. What is the linear approximation for any generic linear function [latex]y=mx+b[/latex]?
  2. Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result.
  3. Explain why the linear approximation becomes less accurate as you increase the distance between [latex]x[/latex] and [latex]a[/latex]. Use a graph to prove your argument.
  4. When is the linear approximation exact?

For the following exercises (5-7), find the linear approximation [latex]L(x)[/latex] to [latex]y=f(x)[/latex] near [latex]x=a[/latex] for the function.

  1. [latex]f(x)=\frac{1}{x}, \, a=2[/latex]
  2. [latex]f(x)= \sin x, \, a=\frac{\pi }{2}[/latex]
  3. [latex]f(x)= \sin^2 x, \, a=0[/latex]

For the following exercises (8-10), compute the values given within 0.01 by deciding on the appropriate [latex]f(x)[/latex] and [latex]a[/latex], and evaluating [latex]L(x)=f(a)+f^{\prime}(a)(x-a)[/latex]. Check your answer using a calculator.

  1. [latex]\sin (0.02)[/latex]
  2. [latex](15.99)^{1/4}[/latex]
  3. [latex]\sin (3.14)[/latex]

For the following exercises (11-13), determine the appropriate [latex]f(x)[/latex] and [latex]a[/latex], and evaluate [latex]L(x)=f(a)+f^{\prime}(a)(x-a).[/latex] Calculate the numerical error in the linear approximations that follow.

  1. [latex]\cos (0.01)[/latex]
  2. [latex](1.01)^{-3}[/latex]
  3. [latex]\sqrt{8.99}[/latex]

For the following exercises (14-18), find the differential of the function.

  1. [latex]y=x \cos x[/latex]
  2. [latex]y=\frac{x^2+2}{x-1}[/latex]
  3. [latex]y=\frac{1}{x+1}[/latex], [latex]x=1[/latex], [latex]dx=0.25[/latex]
  4. [latex]y=\frac{3x^2+2}{\sqrt{x+1}}[/latex], [latex]x=0[/latex], [latex]dx=0.1[/latex]
  5. [latex]y=x^3+2x+\frac{1}{x}[/latex], [latex]x=1[/latex], [latex]dx=0.05[/latex]

For the following exercises (19-21), find the change in volume [latex]dV[/latex] or in surface area [latex]dA[/latex].

  1. [latex]dA[/latex] if the sides of a cube change from [latex]x[/latex] to [latex]x+dx[/latex].
  2. [latex]dV[/latex] if the radius of a sphere changes from [latex]r[/latex] by [latex]dr[/latex].
  3. [latex]dV[/latex] if a circular cylinder of height [latex]3[/latex] changes from [latex]r=2[/latex] to [latex]r=1.9[/latex] cm.

For the following exercise, use differentials to estimate the maximum and relative error when computing the surface area or volume.

  1. A pool has a rectangular base of [latex]10[/latex] ft by [latex]20[/latex] ft and a depth of [latex]6[/latex] ft. What is the change in volume if you only fill it up to [latex]5.5[/latex] ft?

For the following exercises (23-25), confirm the approximations by using the linear approximation at [latex]x=0[/latex].

  1. [latex]\sqrt{1-x}\approx 1-\frac{1}{2}x[/latex]
  2. [latex]\frac{1}{\sqrt{1-x^2}}\approx 1[/latex]
  3. [latex]\sqrt{c^2+x^2}\approx c[/latex]

Maxima and Minima

  1. If you are finding an absolute minimum over an interval [latex][a,b][/latex], why do you need to check the endpoints? Draw a graph that supports your hypothesis.
  2. When you are checking for critical points, explain why you also need to determine points where [latex]f^{\prime}(x)[/latex] is undefined. Draw a graph to support your explanation.
  3. Can you have a finite absolute maximum for [latex]y=ax^3+bx^2+cx+d[/latex] over [latex](−\infty ,\infty )[/latex] assuming [latex]a[/latex] is non-zero? Explain why or why not using graphical arguments.
  4. Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.
  5. Graph the function [latex]y=e^{ax}[/latex]. For which values of [latex]a[/latex], on any infinite domain, will you have an absolute minimum and absolute maximum?

For the following exercises (6-7), determine where the local and absolute maxima and minima occur on the graph given. Assume the graph represents the entirety of each function.

  1. The function graphed starts at (−2.2, 10), decreases rapidly to (−2, −11), increases to (−1, 5) before decreasing slowly to (1, 3), at which point it increases to (2, 7), and then decreases to (3, −20).
  2. The function graphed starts at (−2.5, 1), decreases rapidly to (−2, −1.25), increases to (−1, 0.25) before decreasing slowly to (0, 0.2), at which point it increases slowly to (1, 0.25), then decreases rapidly to (2, −1.25), and finally increases to (2.5, 1).

For the following problems (8-9), draw graphs of [latex]f(x)[/latex], which is continuous, over the interval [latex][-4,4][/latex] with the following properties:

  1. Absolute minimum at [latex]x=1[/latex] and absolute maximum at [latex]x=2[/latex]
  2. Absolute maxima at [latex]x=2[/latex] and [latex]x=-3[/latex], local minimum at [latex]x=1[/latex], and absolute minimum at [latex]x=4[/latex]

For the following exercises (10-14), find the critical points in the domains of the following functions.

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

For the following exercises (15-19), find the local and/or absolute maxima for the functions over the specified domain.

  1. [latex]y=x^2+\frac{2}{x}[/latex] over [latex][1,4][/latex]
  2. [latex]y=\frac{1}{(x-x^2)}[/latex] over [latex](0,1)[/latex]
  3. [latex]y=x+ \sin (x)[/latex] over [latex][0,2\pi][/latex]
  4. [latex]y=|x+1|+|x-1|[/latex] over [latex][-3,2][/latex]
  5. [latex]y= \sin x+ \cos x[/latex] over [latex][0,2\pi][/latex]

For the following exercises (20-22), find the local and absolute minima and maxima for the functions over [latex](−\infty ,\infty )[/latex].

  1. [latex]y=x^2+4x+5[/latex]
  2. [latex]y=3x^4+8x^3-18x^2[/latex]
  3. [latex]y=\dfrac{x^2+x+6}{x-1}[/latex]

For the following functions (23-25), use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.

  1. [latex]y=3x\sqrt{1-x^2}[/latex]
  2. [latex]y=12x^5+45x^4+20x^3-90x^2-120x+3[/latex]
  3. [latex]y=\dfrac{\sqrt{4-x^2}}{\sqrt{4+x^2}}[/latex]

Find the critical points, maxima, and minima for the following piecewise function.

  1. [latex]y=\begin{cases} x^2+1, & x\le 1\\ x^2-4x+5, & x>1 \end{cases}[/latex]

For the following exercise, find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions.

  1. [latex]y=(x-1)^a[/latex], given that [latex]a>1[/latex]

The Mean Value Theorem

  1. Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.
  2. Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.
  3. When are Rolle’s theorem and the Mean Value Theorem equivalent?
  4. If you have a function with a discontinuity, is it still possible to have [latex]f^{\prime}(c)(b-a)=f(b)-f(a)[/latex]? Draw such an example or prove why not.

For the following exercises (5-6), determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

  1. [latex]y=\frac{1}{x^3}[/latex]
  2. [latex]y=\sqrt{x^2-4}[/latex]

For the following exercises (7-8), graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points [latex]c[/latex] such that [latex]f^{\prime}(c)(b-a)=f(b)-f(a)[/latex].

  1. [latex]y=3x^3+2x+1[/latex] over [latex][-1,1][/latex]
  2. [latex]y=x^2 \cos (\pi x)[/latex] over [latex][-2,2][/latex]

For the following exercises (9-11), use the Mean Value Theorem and find all points [latex]0 < c < 2[/latex] such that [latex]f(2)-f(0)=f^{\prime}(c)(2-0)[/latex].

  1. [latex]f(x)=x^3[/latex]
  2. [latex]f(x)= \cos (2\pi x)[/latex]
  3. [latex]f(x)=(x-1)^{10}[/latex]

For the following exercises (12-13), show there is no [latex]c[/latex] such that [latex]f(1)-f(-1)=f^{\prime}(c)(2)[/latex]. Explain why the Mean Value Theorem does not apply over the interval [latex][-1,1][/latex]

  1. [latex]f(x)=|x-\frac{1}{2}|[/latex]
  2. [latex]f(x)=\sqrt{|x|}[/latex]

For the following exercises (14-19), determine whether the Mean Value Theorem applies for the functions over the given interval [latex][a,b][/latex]. Justify your answer.

  1. [latex]y=e^x[/latex] over [latex][0,1][/latex]
  2. [latex]f(x)= \tan (2\pi x)[/latex] over [latex][0,2][/latex]
  3. [latex]y=\dfrac{1}{|x+1|}[/latex] over [latex][0,3][/latex]
  4. [latex]y=\dfrac{x^2+3x+2}{x}[/latex] over [latex][-1,1][/latex]
  5. [latex]y=\ln (x+1)[/latex] over [latex][0,e-1][/latex]
  6. [latex]y=5+|x|[/latex] over [latex][-1,1][/latex]

For the following exercises (20-24), use a calculator to graph the function over the interval [latex][a,b][/latex] and graph the secant line from [latex]a[/latex] to [latex]b[/latex]. Use the calculator to estimate all values of [latex]c[/latex] as guaranteed by the Mean Value Theorem. Then, find the exact value of [latex]c[/latex], if possible, or write the final equation and use a calculator to estimate to four digits.

  1. [latex]y= \tan (\pi x)[/latex] over [latex][-\frac{1}{4},\frac{1}{4}][/latex]
  2. [latex]y=|x^2+2x-4|[/latex] over [latex][-4,0][/latex]
  3. [latex]y=\sqrt{x+1}+\frac{1}{x^2}[/latex] over [latex][3,8][/latex]
  4. Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.
  5. Show that [latex]y= \csc^2 x[/latex] and [latex]y= \cot^2 x[/latex] have the same derivative. What can you say about [latex]y= \csc^2 x - \cot^2 x[/latex]?

Derivatives and the Shape of a Graph

  1. If [latex]c[/latex] is a critical point of [latex]f(x)[/latex], when is there no local maximum or minimum at [latex]c[/latex]? Explain.
  2. For the function [latex]y=x^3[/latex], is [latex]x=0[/latex] both an inflection point and a local maximum/minimum?
  3. For the function [latex]y=x^3[/latex], is [latex]x=0[/latex] an inflection point?
  4. Is it possible for a point [latex]c[/latex] to be both an inflection point and a local extrema of a twice differentiable function?
  5. Why do you need continuity for the first derivative test? Come up with an example.
  6. Explain whether a concave-down function has to cross [latex]y=0[/latex] for some value of [latex]x[/latex].
  7. Explain whether a polynomial of degree [latex]2[/latex] can have an inflection point.

For the following exercises (8-10), analyze the graphs of [latex]f^{\prime}[/latex], then list all intervals where [latex]f[/latex] is increasing or decreasing.

  1. The function f’(x) is graphed. The function starts negative and crosses the x axis at (−2, 0). Then it continues increasing a little before decreasing and crossing the x axis at (−1, 0). It achieves a local minimum at (1, −6) before increasing and crossing the x axis at (2, 0).
  2. The function f’(x) is graphed. The function starts negative and touches the x axis at the origin. Then it decreases a little before increasing to cross the x axis at (1, 0) and continuing to increase.
  3.  The function f’(x) is graphed. The function starts at (−2, 0), decreases to (−1.5, −1.5), increases to (−1, 0), and continues increasing before decreasing to the origin. Then the other side is symmetric: that is, the function increases and then decreases to pass through (1, 0). It continues decreasing to (1.5, −1.5), and then increase to (2, 0).

For the following exercises (11-12), analyze the graphs of [latex]f^{\prime}[/latex], then list

  1. all intervals where [latex]f[/latex] is increasing and decreasing and
  2. where the minima and maxima are located.
  1.  The function f’(x) is graphed. The function starts at (−2, 0), increases and then decreases to (−1, 0), decreases and then increases to an inflection point at the origin. Then the function increases and decreases to cross (1, 0). It continues decreasing and then increases to (2, 0).
  2. The function f’(x) is graphed. The function starts negative and crosses the x axis at the origin, which is an inflection point. Then it continues increasing.

For the following exercises (13-15), analyze the graphs of [latex]f^{\prime}[/latex], then list all inflection points and intervals [latex]f[/latex] that are concave up and concave down.

  1.  The function f’(x) is graphed. The function is linear and starts negative. It crosses the x axis at the origin.
  2.  The function f’(x) is graphed. The function resembles the graph of x3: that is, it starts negative and crosses the x axis at the origin. Then it continues increasing.
  3. The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing to a local maximum at (0, 1), at which point it decreases and touches the x axis at (1, 0). It then increases.

For the following exercises (16-17), draw a graph that satisfies the given specifications for the domain [latex]x=[-3,3][/latex]. The function does not have to be continuous or differentiable.

  1. [latex]f^{\prime}(x)>0[/latex] over [latex]x>2, \, -3
  2. There is a local maximum at [latex]x=2[/latex], local minimum at [latex]x=1[/latex], and the graph is neither concave up nor concave down.

For the following exercise, determine,

  1. intervals where [latex]f[/latex] is concave up or concave down
  2. the inflection points of [latex]f[/latex].
  1. [latex]f(x)=x^3-4x^2+x+2[/latex]

For the following exercises (19-21), determine

  1. intervals where [latex]f[/latex] is increasing or decreasing,
  2. local minima and maxima of [latex]f[/latex],
  3. intervals where [latex]f[/latex] is concave up and concave down, and
  4. the inflection points of [latex]f[/latex].
  1. [latex]f(x)=x^3-6x^2[/latex]
  2. [latex]f(x)=x^{11}-6x^{10}[/latex]
  3. [latex]f(x)=x^2+x+1[/latex]

For the following exercises (22-26), determine,

  1. intervals where [latex]f[/latex] is increasing or decreasing,
  2. local minima and maxima of [latex]f[/latex],
  3. intervals where [latex]f[/latex] is concave up and concave down, and
  4. the inflection points of [latex]f[/latex]. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.
  1. [latex]f(x)= \sin (\pi x)- \cos (\pi x)[/latex] over [latex]x=[-1,1][/latex]
  2. [latex]f(x)= \sin x+ \tan x[/latex] over [latex]\left(-\frac{\pi }{2},\frac{\pi }{2}\right)[/latex]
  3. [latex]f(x)=\dfrac{1}{1-x}, \, x \ne 1[/latex]
  4. [latex]f(x)= \sin x e^x[/latex] over [latex]x=[−\pi ,\pi][/latex]
  5. [latex]f(x)=\frac{1}{4}\sqrt{x}+\frac{1}{x}, \, x>0[/latex]

For the following exercises (27-29), interpret the sentences in terms of [latex]f, \, f^{\prime}[/latex], and [latex]f^{\prime \prime}[/latex].

  1. The population is growing more slowly. Here [latex]f[/latex] is the population. [
  2. The airplane lands smoothly. Here [latex]f[/latex] is the plane’s altitude.
  3. The economy is picking up speed. Here [latex]f[/latex] is a measure of the economy, such as GDP.

For the following exercises (30-31), consider a third-degree polynomial [latex]f(x)[/latex], which has the properties [latex]f^{\prime}(1)=0, \, f^{\prime}(3)=0[/latex]. Determine whether the following statements are true or false. Justify your answer.

  1. [latex]f^{\prime \prime}(x)=0[/latex] for some [latex]1 \le x \le 3[/latex]
  2. If [latex]f(x)[/latex] has three roots, then it has [latex]1[/latex] inflection point.