Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Chapter 10.4, Problem 19E
To determine
To find: The equation of the parabola with its vertex at origin, opens rightward and has directrix
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6.
(i)
Sketch the trace of the following curve on R²,
(t) = (sin(t), 3 sin(t)),
tЄ [0, π].
[3 Marks]
Total marks 10
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Find the length of this curve.
[7 Marks]
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Chapter 10 Solutions
Calculus: Early Transcendentals (2nd Edition)
Ch. 10.1 - Explain how a pair of parametric equations...Ch. 10.1 - Prob. 2ECh. 10.1 - Prob. 3ECh. 10.1 - Give parametric equations that generate the line...Ch. 10.1 - Prob. 5ECh. 10.1 - Prob. 6ECh. 10.1 - Prob. 7ECh. 10.1 - Prob. 8ECh. 10.1 - Prob. 9ECh. 10.1 - Explain how to find points on the curve x = f(t),...
Ch. 10.1 - Prob. 11ECh. 10.1 - Prob. 12ECh. 10.1 - Prob. 13ECh. 10.1 - Prob. 14ECh. 10.1 - Prob. 15ECh. 10.1 - Prob. 16ECh. 10.1 - Prob. 17ECh. 10.1 - Prob. 18ECh. 10.1 - Prob. 19ECh. 10.1 - Prob. 20ECh. 10.1 - Prob. 21ECh. 10.1 - Prob. 22ECh. 10.1 - Prob. 23ECh. 10.1 - Prob. 24ECh. 10.1 - Prob. 25ECh. 10.1 - Prob. 26ECh. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Prob. 30ECh. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Prob. 32ECh. 10.1 - Prob. 33ECh. 10.1 - Prob. 34ECh. 10.1 - Prob. 35ECh. 10.1 - Prob. 36ECh. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Prob. 40ECh. 10.1 - Prob. 41ECh. 10.1 - Prob. 42ECh. 10.1 - Prob. 43ECh. 10.1 - Prob. 44ECh. 10.1 - Curves to parametric equations Give a set of...Ch. 10.1 - Curves to parametric equations Give a set of...Ch. 10.1 - Prob. 47ECh. 10.1 - Prob. 48ECh. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - Prob. 55ECh. 10.1 - Beautiful curves Consider the family of curves...Ch. 10.1 - Prob. 57ECh. 10.1 - Prob. 58ECh. 10.1 - Prob. 59ECh. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Prob. 62ECh. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Prob. 64ECh. 10.1 - Explain why or why not Determine whether the...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Prob. 70ECh. 10.1 - Prob. 71ECh. 10.1 - Prob. 72ECh. 10.1 - Prob. 73ECh. 10.1 - Prob. 74ECh. 10.1 - Prob. 75ECh. 10.1 - Prob. 76ECh. 10.1 - Prob. 77ECh. 10.1 - Prob. 78ECh. 10.1 - Prob. 79ECh. 10.1 - Prob. 80ECh. 10.1 - Prob. 81ECh. 10.1 - Prob. 82ECh. 10.1 - Eliminating the parameter Eliminate the parameter...Ch. 10.1 - Eliminating the parameter Eliminate the parameter...Ch. 10.1 - Prob. 85ECh. 10.1 - Prob. 86ECh. 10.1 - Prob. 87ECh. 10.1 - Prob. 88ECh. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Prob. 93ECh. 10.1 - Prob. 94ECh. 10.1 - Prob. 95ECh. 10.1 - Lissajous curves Consider the following Lissajous...Ch. 10.1 - Lam curves The Lam curve described by...Ch. 10.1 - Prob. 98ECh. 10.1 - Prob. 99ECh. 10.1 - Prob. 100ECh. 10.1 - Prob. 101ECh. 10.1 - Prob. 102ECh. 10.1 - Prob. 103ECh. 10.1 - Air drop A plane traveling horizontally at 80 m/s...Ch. 10.1 - Air dropinverse problem A plane traveling...Ch. 10.1 - Prob. 106ECh. 10.1 - Implicit function graph Explain and carry out a...Ch. 10.1 - Prob. 108ECh. 10.1 - Prob. 109ECh. 10.1 - Prob. 110ECh. 10.2 - Plot the points with polar coordinates (2,6) and...Ch. 10.2 - Prob. 2ECh. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - What is the polar equation of the vertical line x...Ch. 10.2 - What is the polar equation of the horizontal line...Ch. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.2 - Graph the points with the following polar...Ch. 10.2 - Graph the points with the following polar...Ch. 10.2 - Prob. 11ECh. 10.2 - Prob. 12ECh. 10.2 - Prob. 13ECh. 10.2 - Points in polar coordinates Give two sets of polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Prob. 27ECh. 10.2 - Prob. 28ECh. 10.2 - Prob. 29ECh. 10.2 - Prob. 30ECh. 10.2 - Prob. 31ECh. 10.2 - Prob. 32ECh. 10.2 - Prob. 33ECh. 10.2 - Prob. 34ECh. 10.2 - Prob. 35ECh. 10.2 - Prob. 36ECh. 10.2 - Prob. 37ECh. 10.2 - Prob. 38ECh. 10.2 - Prob. 39ECh. 10.2 - Prob. 40ECh. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Prob. 43ECh. 10.2 - Prob. 44ECh. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Prob. 49ECh. 10.2 - Prob. 50ECh. 10.2 - Prob. 51ECh. 10.2 - Prob. 52ECh. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Prob. 55ECh. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Prob. 59ECh. 10.2 - Prob. 60ECh. 10.2 - Prob. 61ECh. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Prob. 66ECh. 10.2 - Prob. 67ECh. 10.2 - Prob. 68ECh. 10.2 - Prob. 69ECh. 10.2 - Prob. 70ECh. 10.2 - Prob. 71ECh. 10.2 - Prob. 72ECh. 10.2 - Prob. 73ECh. 10.2 - Prob. 74ECh. 10.2 - Circles in general Show that the polar equation...Ch. 10.2 - Prob. 76ECh. 10.2 - Prob. 77ECh. 10.2 - Prob. 78ECh. 10.2 - Prob. 79ECh. 10.2 - Prob. 80ECh. 10.2 - Prob. 81ECh. 10.2 - Equations of circles Find equations of the circles...Ch. 10.2 - Prob. 83ECh. 10.2 - Prob. 84ECh. 10.2 - Prob. 85ECh. 10.2 - Prob. 86ECh. 10.2 - Prob. 87ECh. 10.2 - Prob. 88ECh. 10.2 - Prob. 89ECh. 10.2 - Limiting limaon Consider the family of limaons r =...Ch. 10.2 - Prob. 91ECh. 10.2 - Prob. 92ECh. 10.2 - Prob. 93ECh. 10.2 - The lemniscate family Equations of the form r2 = a...Ch. 10.2 - The rose family Equations of the form r = a sin m...Ch. 10.2 - Prob. 96ECh. 10.2 - Prob. 97ECh. 10.2 - The rose family Equations of the form r = a sin m...Ch. 10.2 - Prob. 99ECh. 10.2 - Prob. 100ECh. 10.2 - Prob. 101ECh. 10.2 - Spirals Graph the following spirals. Indicate the...Ch. 10.2 - Prob. 103ECh. 10.2 - Prob. 104ECh. 10.2 - Prob. 105ECh. 10.2 - Prob. 106ECh. 10.2 - Enhanced butterfly curve The butterfly curve of...Ch. 10.2 - Prob. 108ECh. 10.2 - Prob. 109ECh. 10.2 - Prob. 110ECh. 10.2 - Prob. 111ECh. 10.2 - Cartesian lemniscate Find the equation in...Ch. 10.2 - Prob. 113ECh. 10.2 - Prob. 114ECh. 10.3 - Prob. 1ECh. 10.3 - Prob. 2ECh. 10.3 - Explain why the slope of the line tangent to the...Ch. 10.3 - What integral must be evaluated to find the area...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Prob. 18ECh. 10.3 - Prob. 19ECh. 10.3 - Prob. 20ECh. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Prob. 37ECh. 10.3 - Prob. 38ECh. 10.3 - Prob. 39ECh. 10.3 - Prob. 40ECh. 10.3 - Prob. 41ECh. 10.3 - Prob. 42ECh. 10.3 - Prob. 43ECh. 10.3 - Prob. 44ECh. 10.3 - Prob. 45ECh. 10.3 - Multiple identities Explain why the point (1, 3/2)...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Prob. 51ECh. 10.3 - Prob. 52ECh. 10.3 - Regions bounded by a spiral Let Rn be the region...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Prob. 57ECh. 10.3 - Prob. 58ECh. 10.3 - Grazing goat problems Consider the following...Ch. 10.3 - Grazing goat problems Consider the following...Ch. 10.3 - Prob. 61ECh. 10.3 - Tangents and normals Let a polar curve be...Ch. 10.3 - Prob. 63ECh. 10.4 - Give the property that defines all parabolas.Ch. 10.4 - Prob. 2ECh. 10.4 - Give the property that defines all hyperbolas.Ch. 10.4 - Prob. 4ECh. 10.4 - Prob. 5ECh. 10.4 - What is the equation of the standard parabola with...Ch. 10.4 - Prob. 7ECh. 10.4 - Prob. 8ECh. 10.4 - Given vertices (a, 0) and eccentricity e, what are...Ch. 10.4 - Prob. 10ECh. 10.4 - What are the equations of the asymptotes of a...Ch. 10.4 - Prob. 12ECh. 10.4 - Graphing parabolas Sketch a graph of the following...Ch. 10.4 - Prob. 14ECh. 10.4 - Prob. 15ECh. 10.4 - Prob. 16ECh. 10.4 - Prob. 17ECh. 10.4 - Graphing parabolas Sketch a graph of the following...Ch. 10.4 - Prob. 19ECh. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - Prob. 22ECh. 10.4 - Prob. 23ECh. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - Prob. 27ECh. 10.4 - Prob. 28ECh. 10.4 - Prob. 29ECh. 10.4 - Prob. 30ECh. 10.4 - Prob. 31ECh. 10.4 - Prob. 32ECh. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Prob. 36ECh. 10.4 - Prob. 37ECh. 10.4 - Prob. 38ECh. 10.4 - Prob. 39ECh. 10.4 - Prob. 40ECh. 10.4 - Prob. 41ECh. 10.4 - Prob. 42ECh. 10.4 - Prob. 43ECh. 10.4 - Prob. 44ECh. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Prob. 48ECh. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Prob. 55ECh. 10.4 - Prob. 56ECh. 10.4 - Prob. 57ECh. 10.4 - Prob. 58ECh. 10.4 - Prob. 59ECh. 10.4 - Prob. 60ECh. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Prob. 65ECh. 10.4 - Hyperbolas with a graphing utility Use a graphing...Ch. 10.4 - Prob. 67ECh. 10.4 - Prob. 68ECh. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Prob. 72ECh. 10.4 - Prob. 73ECh. 10.4 - Prob. 74ECh. 10.4 - Prob. 75ECh. 10.4 - The ellipse and the parabola Let R be the region...Ch. 10.4 - Tangent lines for an ellipse Show that an equation...Ch. 10.4 - Prob. 78ECh. 10.4 - Volume of an ellipsoid Suppose that the ellipse...Ch. 10.4 - Area of a sector of a hyperbola Consider the...Ch. 10.4 - Volume of a hyperbolic cap Consider the region R...Ch. 10.4 - Prob. 82ECh. 10.4 - Prob. 83ECh. 10.4 - Golden Gate Bridge Completed in 1937, San...Ch. 10.4 - Prob. 85ECh. 10.4 - Prob. 86ECh. 10.4 - Prob. 87ECh. 10.4 - Prob. 88ECh. 10.4 - Shared asymptotes Suppose that two hyperbolas with...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Prob. 93ECh. 10.4 - Prob. 94ECh. 10.4 - Confocal ellipse and hyperbola Show that an...Ch. 10.4 - Approach to asymptotes Show that the vertical...Ch. 10.4 - Prob. 97ECh. 10.4 - Prob. 98ECh. 10.4 - Prob. 99ECh. 10 - Explain why or why not Determine whether the...Ch. 10 - Prob. 2RECh. 10 - Prob. 3RECh. 10 - Prob. 4RECh. 10 - Prob. 5RECh. 10 - Prob. 6RECh. 10 - Prob. 7RECh. 10 - Prob. 8RECh. 10 - Eliminating the parameter Eliminate the parameter...Ch. 10 - Prob. 10RECh. 10 - Parametric description Write parametric equations...Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Prob. 13RECh. 10 - Prob. 14RECh. 10 - Parametric description Write parametric equations...Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 18RECh. 10 - Prob. 19RECh. 10 - Sets in polar coordinates Sketch the following...Ch. 10 - Prob. 21RECh. 10 - Prob. 22RECh. 10 - Polar conversion Write the equation...Ch. 10 - Polar conversion Consider the equation r = 4/(sin ...Ch. 10 - Prob. 25RECh. 10 - Prob. 26RECh. 10 - Prob. 27RECh. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Prob. 31RECh. 10 - The region enclosed by all the leaves of the rose...Ch. 10 - The region enclosed by the limaon r = 3 cosCh. 10 - The region inside the limaon r = 2 + cos and...Ch. 10 - Prob. 35RECh. 10 - Prob. 36RECh. 10 - The area that is inside the cardioid r = 1 + cos ...Ch. 10 - Prob. 38RECh. 10 - Prob. 39RECh. 10 - Prob. 40RECh. 10 - Conic sections a. Determine whether the following...Ch. 10 - Prob. 42RECh. 10 - Prob. 43RECh. 10 - Prob. 44RECh. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 46RECh. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 49RECh. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Prob. 52RECh. 10 - Prob. 53RECh. 10 - Prob. 54RECh. 10 - Eccentricity-directrix approach Find an equation...Ch. 10 - Prob. 56RECh. 10 - Prob. 57RECh. 10 - Prob. 58RECh. 10 - Prob. 59RECh. 10 - Prob. 60RECh. 10 - Prob. 61RECh. 10 - Prob. 62RECh. 10 - Prob. 63RECh. 10 - Prob. 64RECh. 10 - Prob. 65RECh. 10 - Prob. 66RECh. 10 - Prob. 67RECh. 10 - Prob. 68RECh. 10 - Prob. 69RECh. 10 - Prob. 70RECh. 10 - Prob. 71RE
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- Total marks 15 Total marks on paper: 80 6. Let DCR2 be a bounded domain with the boundary ǝD which can be represented as a smooth closed curve : [a, b] → R², oriented in the anticlockwise direction. (i) Use Green's Theorem to justify that the area of the domain D can be computed by the formula 1 Area(D) = . [5 Marks] (ii) Use the area formula in (i) to find the area of the domain D enclosed by the ellipse (t) = (5 cos(t), 10 sin(t)), t = [0,2π]. [5 Marks] (iii) Explain in your own words why Green's Theorem can not be applied to the vector field У x F(x,y) = ( - x² + y²²x² + y² ). [5 Marks]arrow_forwardTotal marks 15 པ་ (i) Sketch the trace of the following curve on R2, (t) = (t2 cos(t), t² sin(t)), t = [0,2π]. [3 Marks] (ii) Find the length of this curve. (iii) [7 Marks] Give a parametric representation of a curve : [0, that has initial point (1,0), final point (0, 1) and the length √2. → R² [5 Marks] Turn over. MA-201: Page 4 of 5arrow_forwardTotal marks 15 5. (i) Let f R2 R be defined by f(x1, x2) = x² - 4x1x2 + 2x3. Find all local minima of f on R². (ii) [10 Marks] Give an example of a function f: R2 R which is not bounded above and has exactly one critical point, which is a minimum. Justify briefly your answer. [5 Marks] 6. (i) Sketch the trace of the following curve on R2, y(t) = (sin(t), 3 sin(t)), t = [0,π]. [3 Marks]arrow_forward
- A ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder, its x coordinate, at time t seconds is given by x(t)=7+2t. wall y(1) 25 ft. ladder x(1) ground (a) Find the formula for the location of the top of the ladder, the y coordinate, as a function of time t. The formula for y(t)= √ 25² - (7+2t)² (b) The domain of t values for y(t) ranges from 0 (c) Calculate the average velocity of the top of the ladder on each of these time intervals (correct to three decimal places): . (Put your cursor in the box, click and a palette will come up to help you enter your symbolic answer.) time interval ave velocity [0,2] -0.766 [6,8] -3.225 time interval ave velocity -1.224 -9.798 [2,4] [8,9] (d) Find a time interval [a,9] so that the average velocity of the top of the ladder on this…arrow_forwardTotal marks 15 3. (i) Let FRN Rm be a mapping and x = RN is a given point. Which of the following statements are true? Construct counterex- amples for any that are false. (a) If F is continuous at x then F is differentiable at x. (b) If F is differentiable at x then F is continuous at x. If F is differentiable at x then F has all 1st order partial (c) derivatives at x. (d) If all 1st order partial derivatives of F exist and are con- tinuous on RN then F is differentiable at x. [5 Marks] (ii) Let mappings F= (F1, F2) R³ → R² and G=(G1, G2) R² → R² : be defined by F₁ (x1, x2, x3) = x1 + x², G1(1, 2) = 31, F2(x1, x2, x3) = x² + x3, G2(1, 2)=sin(1+ y2). By using the chain rule, calculate the Jacobian matrix of the mapping GoF R3 R², i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)? (iii) [7 Marks] Give reasons why the mapping Go F is differentiable at (0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0). [3 Marks]arrow_forward5. (i) Let f R2 R be defined by f(x1, x2) = x² - 4x1x2 + 2x3. Find all local minima of f on R². (ii) [10 Marks] Give an example of a function f: R2 R which is not bounded above and has exactly one critical point, which is a minimum. Justify briefly Total marks 15 your answer. [5 Marks]arrow_forward
- Total marks 15 4. : Let f R2 R be defined by f(x1, x2) = 2x²- 8x1x2+4x+2. Find all local minima of f on R². [10 Marks] (ii) Give an example of a function f R2 R which is neither bounded below nor bounded above, and has no critical point. Justify briefly your answer. [5 Marks]arrow_forward4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 Marks]arrow_forward4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, so that ||hn||| 0. Thus we find ||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)|| (**) ||DF(x)hn|| + ||R(hn) || → 0, because the linear mapping DF(x) is continuous and for all large n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]arrow_forward
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