can remember the formula for the slope of the tangent line by
thinking that symbolically, the dts cancel, leaving you
with dy ⁄ dx.
find the equation of the tangent line, you also need a point. Use
the given value of t into x(t) and y(t)
to find it. Then you can use the point-slope formula from high
school algebra (y − y0 = m(x − x0
)) to find the equation.
you arent given a value of t, but the coordinates (x,
y) instead. Then you must find which value of t gives
you the correct (x(t), y(t)). Make sure
you check that both x and y are correct for your value
can remember the arclength formula by recalling that it is derived
from the distance formula between two points, which in turn comes
from the Pythagorean Theorem.
make the common algebraic mistake of thinking that
reduces to a + b! This is extremely wrong, and your
teacher will likely be merciless if you do it
problems in Calculus II classes are rigged so that when you
expand x′(t)² + y′(t)²
, it becomes a perfect square that cancels nicely with the square
this perfect square is achieved by making one of x′(t)²
and y′(t)² be something of the form (a −
b)² = a² − 2ab + b².
Then the other one changes it to a² + 2ab + b²,
which you can then factor as (a + b)².
common technique in arclength problems is to make a u-substitution
for whatever is under the square root sign. Then (hopefully) you can
manipulate the expression outside the square root into being the du.
However, you might have to do several steps of algebraic
manipulation, pulling factors in or out of the square root sign,
before this works.
may sometimes be able to use symmetry to find the arclength of part
of a curve and then multiply by an appropriate factor to get the
total arclength. This can be especially helpful if you just want to
examine part of the curve where all the quantities involved are
its feasible, check that your answer makes sense. Unlike area
integrals, which can be negative if a curve goes below the x-axis,
arclength should always be positive! You might also be able to check
that the curve looks about as long as your answer.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.