How do you find the equation of the tangent to the curve #x=t^4+1#, #y=t^3+t# at the point where #t=-1# ?
Shop Grip-Rite PrimeGuard Plus 9 x 2-1/2-in Polymer Deck Screws (5-lb) in the Deck Screws department at Lowe's.com. PGP exterior screws are used for exterior projects like decks or fences and used in wood to wood applications. Top Choice 2-in x 4-in x 16-ft Southern Yellow Pine Lumber. Piccollage 1 1. Compare; Find My Store. For pricing and availability. Top Choice 2-in x 4-in x 14-ft Spruce Pine Fir Lumber. Model #24#2SE.14. Compare; Find My Store. For pricing and availability. Top Choice 2-in x 4-in x 10-ft Douglas Fir Lumber.
Get step-by-step answers and hints for your math homework problems. Learn the basics, check your work, gain insight on different ways to solve problems. For chemistry, calculus, algebra, trigonometry, equation solving, basic math and more. Some of the most reviewed Plexiglas acrylic sheets are the Plexiglas 24 in. Acrylic Sheet (4-Pack) with 45 reviews and the Plexiglas 48 in. Acrylic sheet with 29 reviews.
Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music Wolfram Alpha brings expert-level knowledge.
1 Answer
To find the equation of the tangent line, we need the slope
#m = dy/dx# and the point of tangency #(x_o,y_o)# .
Then the equation is the usual # y-y_o = m(x - x_o).#
We have the parametric curve #x = t^4+1, y = t^3+t# ,
so we compute #dx/dt = 4t^3 and dy/dt = 3t^2+1.#
The chain rule #dy/dt = dy/dx*dx/dt# says that #dy/dx = (dy/dt)/(dx/dt).#
So we use the derivatives of the parametric equations:
Also at #t=-1# the original equations give
#(x_o,y_o) = ((-1)^4+1,(-1)^3+(-1))=(1+1,(-1)+(-1))=(2,-2)#
Now we put in the info for the tangent line:
#y-y_o=m(x-x_o)#
#y-(-2)=(-1)(x-2)# or
#y+2=-x+2# or just plain old #y=-x# .
Webtolayers 1 1 2 X 4.5
Another great answer from the modest dansmath! /
Related topic
Related questions
Webtolayers 1 1 2 X 47
First six summands drawn as portions of a square.
The geometric series on the real line.
In mathematics, the infinite series1/2 + 1/4 + 1/8 + 1/16 + ··· is an elementary example of a geometric series that converges absolutely.
There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2−1 + 2−2 + 2−3 + .. Sigpro 2 1.
The sum of this series can be denoted in summation notation as:
Proof[edit]
Webtolayers 1 1 2 X 4 1 2
As with any infinite series, the infinite sum
is defined to mean the limit of the sum of the first n terms
as n approaches infinity.
Multiplying sn by 2 reveals a useful relationship:
Subtracting sn from both sides,
History[edit]
Zeno's paradox[edit]
Webtolayers 1 1 2 X 42
This series was used as a representation of many of Zeno's paradoxes, one of which, Achilles and the Tortoise, is shown here.[1] In the paradox, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but by then, the tortoise would already have moved another five meters. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on. Zeno argued that the tortoise would always remain ahead of Achilles.
The Eye of Horus[edit]
The parts of the Eye of Horus were once thought to represent the first six summands of the series.[2]
In a myriad ages it will not be exhausted[edit]
'Zhuangzi', also known as 'South China Classic', written by Zhuang Zhou. In the miscellaneous chapters 'All Under Heaven', he said: 'Take a chi long stick and remove half every day, in a myriad ages it will not be exhausted.'
See also[edit]
References[edit]
- ^Wachsmuth, Bet G. 'Description of Zeno's paradoxes'. Archived from the original on 2014-12-31. Retrieved 2014-12-29.
- ^Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN978 1 84668 292 6.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_⋯&oldid=983537481'