Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
3rd Edition
ISBN: 9780134995991
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Chapter 14.2, Problem 11E
To determine
To find: The derivative of the
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Question 1. (10 points)
A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by
dV
=
1.45V(2 In(V+1)).
dt
(a) (4 pts) Find all the equilibria and determine their stability using the stability condition.
(b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of
f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable.
(c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain
in biological terms what happens to the size of each of these tumours at time progresses.
For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.
Question 2: (10 points) Evaluate the definite integral.
Use the following form of the definition of the integral to evaluate the integral:
Theorem: Iff is integrable on [a, b], then
where Ax = (ba)/n and x₂ = a + i^x.
You might need the following formulas.
IM³
L² (3x²
(3x²+2x-
2x - 1)dx.
n
[f(z)dz lim f(x)Az
a
n→∞
i=1
n(n + 1)
2
n
i=1
n(n+1)(2n+1)
6
Chapter 14 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
Ch. 14.1 - Restrict the domain o f the vector function in...Ch. 14.1 - Explain why the curve in Example 5 lies on the...Ch. 14.1 - How many independent variables does the function...Ch. 14.1 - How many dependent scalar variables does the...Ch. 14.1 - Prob. 3ECh. 14.1 - Prob. 4ECh. 14.1 - How do you evaluate limtar(t), where r(t) = f(t),...Ch. 14.1 - How do you determine whether r(t) = f(t) i + g(t)...Ch. 14.1 - Find a function r(t) for the line passing through...Ch. 14.1 - Find a function r(t) whose graph is a circle of...
Ch. 14.1 - Prob. 9ECh. 14.1 - Prob. 10ECh. 14.1 - Lines and line segments Find a function r(t) that...Ch. 14.1 - 914. Lines and line segments Find a function r(t)...Ch. 14.1 - Prob. 13ECh. 14.1 - Prob. 14ECh. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Limits Evaluate the following limits. 41....Ch. 14.1 - Limits Evaluate the following limits. 42....Ch. 14.1 - Limits Evaluate the following limits. 43....Ch. 14.1 - Limits Evaluate the following limits. 44....Ch. 14.1 - Limits Evaluate the following limits. 45....Ch. 14.1 - Limits Evaluate the following limits. 46....Ch. 14.1 - Prob. 37ECh. 14.1 - Domains Find the domain of the following...Ch. 14.1 - Domains Find the domain of the following...Ch. 14.1 - Domains Find the domain of the following...Ch. 14.1 - Prob. 41ECh. 14.1 - Curve-plane intersections Find the points (if they...Ch. 14.1 - Curve-plane intersections Find the points (if they...Ch. 14.1 - Curve-plane intersections Find the points (if they...Ch. 14.1 - Matching functions with graphs Match functions af...Ch. 14.1 - Prob. 46ECh. 14.1 - 4750. Curve of intersection Find a function r(t)...Ch. 14.1 - 4750. Curve of intersection Find a function r(t)...Ch. 14.1 - 4750. Curve of intersection Find a function r(t)...Ch. 14.1 - Curve of intersection Find a function r(t) that...Ch. 14.1 - Golf slice A golfer launches a tee shot down a...Ch. 14.1 - Curves on surfaces Verify that the curve r(t) lies...Ch. 14.1 - 5256. Curves on surfaces Verify that the curve...Ch. 14.1 - Curves on surfaces Verify that the curve r(t) lies...Ch. 14.1 - Curves on surfaces Verify that the curve r(t) lies...Ch. 14.1 - 5256. Curves on surfaces Verify that the curve...Ch. 14.1 - 5758. Closest point on a curve Find the point P on...Ch. 14.1 - 5758. Closest point on a curve Find the point P on...Ch. 14.1 - Curves on spheres 75. Graph the curve...Ch. 14.1 - Prob. 60ECh. 14.1 - Prob. 61ECh. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Limits of vector functions Let r(t) = (f(t), g(t),...Ch. 14.2 - Prob. 1QCCh. 14.2 - Suppose r(t) has units of m/s. Explain why T(t) =...Ch. 14.2 - Let u(t)=t,t,t and v(t)=1,1,1 compute...Ch. 14.2 - Let r(t)=1,2t,3t2. Compute r(t)dt.Ch. 14.2 - Prob. 1ECh. 14.2 - Explain the geometric meaning of r(t).Ch. 14.2 - Prob. 3ECh. 14.2 - Compute r(t) when r(t) = t10, 8t, cos t.Ch. 14.2 - How do you find the indefinite integral of r(t) =...Ch. 14.2 - How do you evaluate abr(t)dt?Ch. 14.2 - Find C if r(t)=et,3cost,t+10+C and r(0)=0,0,0.Ch. 14.2 - Find the unit tangent vector at t = 0 for the...Ch. 14.2 - Derivatives of vector-valued functions...Ch. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Derivatives of vector-valued functions...Ch. 14.2 - Prob. 13ECh. 14.2 - Derivatives of vector-valued functions...Ch. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Prob. 17ECh. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Prob. 20ECh. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Prob. 26ECh. 14.2 - Prob. 27ECh. 14.2 - Prob. 28ECh. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Prob. 39ECh. 14.2 - Prob. 40ECh. 14.2 - Prob. 41ECh. 14.2 - Derivative rules Suppose u and v are...Ch. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Prob. 44ECh. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Prob. 46ECh. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Prob. 54ECh. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Prob. 60ECh. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Prob. 66ECh. 14.2 - Prob. 67ECh. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Prob. 79ECh. 14.2 - Prob. 80ECh. 14.2 - Prob. 81ECh. 14.2 - Prob. 82ECh. 14.2 - Prob. 83ECh. 14.2 - Relationship between r and r 78. Consider the...Ch. 14.2 - Relationship between r and r 79. Consider the...Ch. 14.2 - Prob. 86ECh. 14.2 - Relationship between r and r 81. Consider the...Ch. 14.2 - Relationship between r and r 82. Consider the...Ch. 14.2 - Relationship between r and r 83. Give two families...Ch. 14.2 - Motion on a sphere Prove that r describes a curve...Ch. 14.2 - Vectors r and r for lines a. If r(t) = at, bt, ct...Ch. 14.2 - Proof of Sum Rule By expressing u and v in terms...Ch. 14.2 - Proof of Product Rule By expressing u in terms of...Ch. 14.2 - Prob. 94ECh. 14.2 - Cusps and noncusps a. Graph the curve r(t) = t3,...Ch. 14.3 - Given r(t)=t,t2,t3, find v(t) and a(t).Ch. 14.3 - Find the functions that give the speed of the two...Ch. 14.3 - Prob. 3QCCh. 14.3 - Prob. 4QCCh. 14.3 - Prob. 5QCCh. 14.3 - Given the position function r of a moving object,...Ch. 14.3 - What is the relationship between the position and...Ch. 14.3 - Write Newtons Second Law of Motion in vector form.Ch. 14.3 - Write Newtons Second Law of Motion for...Ch. 14.3 - Given the acceleration of an object and its...Ch. 14.3 - Given the velocity of an object and its initial...Ch. 14.3 - The velocity of a moving object, for t 0, is...Ch. 14.3 - A baseball is hit 2 feet above home plate, and the...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Prob. 27ECh. 14.3 - Carnival rides 28. Suppose the carnival ride in...Ch. 14.3 - Trajectories on circles and spheres Determine...Ch. 14.3 - Prob. 30ECh. 14.3 - Trajectories on circles and spheres Determine...Ch. 14.3 - Trajectories on circles and spheres Determine...Ch. 14.3 - Path on a sphere show that the following...Ch. 14.3 - Path on a sphere show that the following...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Prob. 50ECh. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Prob. 56ECh. 14.3 - Prob. 57ECh. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Motion on the moon The acceleration due to gravity...Ch. 14.3 - Firing angles A projectile is fired over...Ch. 14.3 - Prob. 64ECh. 14.3 - Speed on an ellipse An object moves along an...Ch. 14.3 - Golf shot A golfer stands 390 ft (130 yd)...Ch. 14.3 - Another golf shot A golfer stands 420 ft (140 yd)...Ch. 14.3 - Prob. 68ECh. 14.3 - Initial speed of a golf shot A golfer stands 420...Ch. 14.3 - Ski jump The lip of a ski jump is 8 m above the...Ch. 14.3 - Designing a baseball pitch A baseball leaves the...Ch. 14.3 - Parabolic trajectories Show that the...Ch. 14.3 - Prob. 73ECh. 14.3 - A race Two people travel from P(4, 0) to Q(4, 0)...Ch. 14.3 - Circular motion Consider an object moving along...Ch. 14.3 - Prob. 76ECh. 14.3 - A circular trajectory An object moves clockwise...Ch. 14.3 - Prob. 78ECh. 14.3 - Tilted ellipse Consider the curve r(t) = cos t,...Ch. 14.3 - Equal area property Consider the ellipse r(t) = a...Ch. 14.3 - Another property of constant | r | motion Suppose...Ch. 14.3 - Prob. 82ECh. 14.3 - Nonuniform straight-line motion Consider the...Ch. 14.4 - What does the arc length formula give for the...Ch. 14.4 - Consider the portion of a circle r(t) = (cos t,...Ch. 14.4 - Prob. 3QCCh. 14.4 - Find the length of the line given by r(t) = t, 2t,...Ch. 14.4 - Explain how to find the length of the curve r(t) =...Ch. 14.4 - Express the arc length of a curve in terms of the...Ch. 14.4 - Suppose an object moves in space with the position...Ch. 14.4 - An object moves on a trajectory given by r(t) = 10...Ch. 14.4 - Use calculus to find the length of the line...Ch. 14.4 - Explain what it means for a curve to be...Ch. 14.4 - Is the curve r(t) = cos t, sin t parameterized by...Ch. 14.4 - Arc length calculations Find the length of he...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Prob. 13ECh. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Prob. 16ECh. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed of Earth Verify that the length of one orbit...Ch. 14.4 - Speed of Jupiter Verify that the length of one...Ch. 14.4 - Arc length approximations Use a calculator to...Ch. 14.4 - Prob. 30ECh. 14.4 - Arc length approximations Use a calculator to...Ch. 14.4 - Prob. 32ECh. 14.4 - Prob. 33ECh. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Prob. 37ECh. 14.4 - Prob. 38ECh. 14.4 - Prob. 39ECh. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Explain why or why not Determine whether the...Ch. 14.4 - Length of a line segment Consider the line segment...Ch. 14.4 - Tilted circles Let the curve C be described by...Ch. 14.4 - Prob. 46ECh. 14.4 - Prob. 47ECh. 14.4 - Toroidal magnetic field A circle of radius a that...Ch. 14.4 - Projectile trajectories A projectile (such as a...Ch. 14.4 - Variable speed on a circle Consider a particle...Ch. 14.4 - Arc length parameterization Prove that the line...Ch. 14.4 - Arc length parameterization Prove that the curve...Ch. 14.4 - Prob. 53ECh. 14.4 - Change of variables Consider the parameterized...Ch. 14.5 - What is the curvature of the circle r() =...Ch. 14.5 - Use the alternative curvature formula to compute...Ch. 14.5 - Prob. 3QCCh. 14.5 - Prob. 4QCCh. 14.5 - Prob. 5QCCh. 14.5 - Prob. 6QCCh. 14.5 - Prob. 7QCCh. 14.5 - What is the curvature of a straight line?Ch. 14.5 - Explain the meaning of the curvature of a curve....Ch. 14.5 - Give a practical formula for computing the...Ch. 14.5 - Interpret the principal unit normal vector of a...Ch. 14.5 - Give a practical formula for computing the...Ch. 14.5 - Explain how to decompose the acceleration vector...Ch. 14.5 - Explain how the vectors T, N, and B are related...Ch. 14.5 - How do you compute B?Ch. 14.5 - Give a geometrical interpretation of the torsion.Ch. 14.5 - How do you compute the torsion?Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Prob. 20ECh. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Prob. 27ECh. 14.5 - Prob. 28ECh. 14.5 - Prob. 29ECh. 14.5 - Prob. 30ECh. 14.5 - Prob. 31ECh. 14.5 - Prob. 32ECh. 14.5 - Prob. 33ECh. 14.5 - Prob. 34ECh. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Prob. 39ECh. 14.5 - Prob. 40ECh. 14.5 - Computing the binormal vector and torsion In...Ch. 14.5 - Computing the binormal vector and torsion In...Ch. 14.5 - Prob. 43ECh. 14.5 - Prob. 44ECh. 14.5 - Prob. 45ECh. 14.5 - Computing the binormal vector and torsion Use the...Ch. 14.5 - Computing the binormal vector and torsion Use the...Ch. 14.5 - Prob. 48ECh. 14.5 - Explain why or why not Determine whether the...Ch. 14.5 - Special formula: Curvature for y = f(x) Assume...Ch. 14.5 - Curvature for y = f(x) Use the result of Exercise...Ch. 14.5 - Prob. 52ECh. 14.5 - Prob. 53ECh. 14.5 - Curvature for y = f(x) Use the result of Exercise...Ch. 14.5 - Prob. 55ECh. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Curvature of ln x Find the curvature of f(x) = ln...Ch. 14.5 - Curvature of ex Find the curvature of f(x) = ex...Ch. 14.5 - Prob. 70ECh. 14.5 - Finding radii of curvature Find the radius of...Ch. 14.5 - Finding radii of curvature Find the radius of...Ch. 14.5 - Finding radii of curvature Find the radius of...Ch. 14.5 - Designing a highway curve The function
r(t) =...Ch. 14.5 - Curvature of the sine curve The function f(x) =...Ch. 14.5 - Parabolic trajectory In Example 7 it was shown...Ch. 14.5 - Parabolic trajectory Consider the parabolic...Ch. 14.5 - Prob. 78ECh. 14.5 - Zero curvature Prove that the curve...Ch. 14.5 - Prob. 80ECh. 14.5 - Maximum curvature Consider the superparabolas...Ch. 14.5 - Alternative derivation of the curvature Derive the...Ch. 14.5 - Computational formula for B Use the result of part...Ch. 14.5 - Prob. 84ECh. 14.5 - Descartes four-circle solution Consider the four...Ch. 14 - Prob. 1RECh. 14 - Sets of points Describe the set of points...Ch. 14 - Graphing curves Sketch the curves described by the...Ch. 14 - Prob. 4RECh. 14 - Curves in space Sketch the curves described by the...Ch. 14 - Curves in space Sketch the curves described by the...Ch. 14 - Intersection curve A sphere S and a plane P...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Prob. 13RECh. 14 - Intersection curve Find the curve r(t) where the...Ch. 14 - Intersection curve Find the curve r(t) where the...Ch. 14 - Prob. 16RECh. 14 - Prob. 17RECh. 14 - Prob. 18RECh. 14 - Prob. 19RECh. 14 - Prob. 20RECh. 14 - Prob. 21RECh. 14 - Prob. 22RECh. 14 - Prob. 23RECh. 14 - Prob. 24RECh. 14 - Finding r from r Find the function r that...Ch. 14 - Finding r from r Find the function r that...Ch. 14 - Prob. 27RECh. 14 - Prob. 28RECh. 14 - Prob. 29RECh. 14 - Velocity and acceleration from position consider...Ch. 14 - Velocity and acceleration from position Consider...Ch. 14 - Solving equations of motion Given an acceleration...Ch. 14 - Prob. 33RECh. 14 - Orthogonal r and r Find all points on the ellipse...Ch. 14 - Modeling motion Consider the motion of the...Ch. 14 - Prob. 36RECh. 14 - Prob. 37RECh. 14 - Firing angles A projectile is fired over...Ch. 14 - Prob. 39RECh. 14 - Baseball motion A toddler on level ground throws a...Ch. 14 - Prob. 41RECh. 14 - Prob. 42RECh. 14 - Prob. 43RECh. 14 - Prob. 44RECh. 14 - Arc length Find the arc length of the following...Ch. 14 - Prob. 46RECh. 14 - Velocity and trajectory length The acceleration of...Ch. 14 - Prob. 48RECh. 14 - Arc length parameterization Find the description...Ch. 14 - Tangents and normals for an ellipse Consider the...Ch. 14 - Prob. 51RECh. 14 - Prob. 52RECh. 14 - Properties of space curves Do the following...Ch. 14 - Prob. 54RECh. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Prob. 59RECh. 14 - Curve analysis Carry out the following steps for...Ch. 14 - Prob. 61RECh. 14 - Prob. 62RECh. 14 - Prob. 63RECh. 14 - Prob. 64RE
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- For the system consisting of the three planes:plane 1: -4x + 4y - 2z = -8plane 2: 2x + 2y + 4z = 20plane 3: -2x - 3y + z = -1a) Are any of the planes parallel and/or coincident? Justify your answer.b) Determine if the normals are coplanar. What does this tell you about the system?c) Solve the system if possible. Show a complete solution (do not use matrix operations). Classify the system using the terms: consistent, inconsistent, dependent and/or independent.arrow_forwardFor the system consisting of the three planes:plane 1: -4x + 4y - 2z = -8plane 2: 2x + 2y + 4z = 20plane 3: -2x - 3y + z = -1a) Are any of the planes parallel and/or coincident? Justify your answer.b) Determine if the normals are coplanar. What does this tell you about the system?c) Solve the system if possible. Show a complete solution (do not use matrix operations). Classify the system using the terms: consistent, inconsistent, dependent and/or independent.arrow_forwardOpen your tool box and find geometric methods, symmetries of even and odd functions and the evaluation theorem. Use these to calculate the following definite integrals. Note that you should not use Riemann sums for this problem. (a) (4 pts) (b) (2 pts) 3 S³ 0 3-x+9-dz x3 + sin(x) x4 + cos(x) dx (c) (4 pts) L 1-|x|dxarrow_forward
- An engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides to do it in three sections. The first section runs from point P to point Q, and costs $48 per mile to lay, the second section runs from point Q to point R and costs $54 per mile, the third runs from point R to point S and costs $44 per mile. Looking at the diagram below, you see that if you know the lengths marked x and y, then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline. Please show your answers to 4 decimal places. 2 Miles x = 1 Mile R 10 miles miles y = milesarrow_forwardAn open-top rectangular box is being constructed to hold a volume of 150 in³. The base of the box is made from a material costing 7 cents/in². The front of the box must be decorated, and will cost 11 cents/in². The remainder of the sides will cost 3 cents/in². Find the dimensions that will minimize the cost of constructing this box. Please show your answers to at least 4 decimal places. Front width: Depth: in. in. Height: in.arrow_forwardFind and classify the critical points of z = (x² – 8x) (y² – 6y). Local maximums: Local minimums: Saddle points: - For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. Enter DNE if there are no points for a classification.arrow_forward
- Suppose that f(x, y, z) = (x − 2)² + (y – 2)² + (z − 2)² with 0 < x, y, z and x+y+z≤ 10. 1. The critical point of f(x, y, z) is at (a, b, c). Then a = b = C = 2. Absolute minimum of f(x, y, z) is and the absolute maximum isarrow_forwardThe spread of an infectious disease is often modeled using the following autonomous differential equation: dI - - BI(N − I) − MI, dt where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of transmission, and μ is the rate at which people recover from infection. Close a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria. b) (5 points) For the equilbria in part a), determine whether each is stable or unstable. c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the dt function by hand.) Identify the equilibria as stable or unstable in the graph. d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.arrow_forwardFind the indefinite integral. Check Answer: 7x 4 + 1x dxarrow_forward
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