Question

1) Consider the curve y = e^cos(x) .

(a) Find y'

(b) Use your answer to part (a) to find the equation of the tangent line to y = e^cos(x) at x = π/2.

2)

3)Consider the curve y = x + 1/x − 1 .

(a) Find y' .

(b) Use your answer to part (a) to find the points on the curve y = x + 1/x − 1 where the tangent line is parallel to the line y = − 1/2 x + 5

Answer #1

1)Consider the curve y = x + 1/x − 1 .
(a) Find y' .
(b) Use your answer to part (a) to find the points on the curve
y = x + 1/x − 1 where the tangent line is parallel to the line y =
− 1/2 x + 5
2) (a) Consider lim h→0 tan^2 (π/3 + h) − 3/h This limit
represents the derivative, f'(a), of some function f at some number
a. State such an...

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

1. Graph the curve given in parametric form by x = e t sin(t)
and y = e t cos(t) on the interval 0 ≤ t ≤ π2.
2. Find the length of the curve in the previous problem.
3. In the polar curve defined by r = 1 − sin(θ) find the points
where the tangent line is vertical.

Find the derivative of the parametric curve x=2t-3t2,
y=cos(3t) for 0 ≤ ? ≤ 2?.
Find the values for t where the tangent lines are horizontal on
the parametric curve. For the horizontal tangent lines, you do not
need to find the (x,y) pairs for these values of t.
Find the values for t where the tangent lines are vertical on
the parametric curve. For these values of t find the coordinates of
the points on the parametric curve.

Find an equation for the line tangent to the following curve at
the point (4,1).
1−y=sin(x+y^(2)−5)
Use symbolic notation and fractions where needed. Express the
equation of the tangent line in terms of y
and x.
equation:

Consider x^2 +sin(y)=4xy^2 +1
a.)Use Implicit differentiation to find dy/dx
b.) find an equation tangent of the line to the curve x^2
+sin(y)=4xy^2 +1 at (1,0)

Consider the parametric curve given by the equations:
x = tsin(t) and y = t cos(t) for 0 ≤ t ≤ 1
(a) Find the slope of a tangent line to this curve when t =
1.
(b) Find the arclength of this curve

Find an equation of the tangent line to the curve at the given
point. A) y = 6x + 3 cos x, P = (0, 3) B)y = 8 x cos x P = \(pi ,
-8 pi)
B)Find an equation of the tangent line to the curve at the given
point.
y = 8 x cos x
C)
If H(θ) = θ cos θ, find H'(θ) and H''(θ).
find H'(
θ)
and H"(θ)

3. Find the equation of the tangent line to the curve 2x^3 + y^2
= xy at the point (−1, 1).
4. Use implicit differentiation to find y' for sin(xy^2 ) − x^3
= 4x + 2y.
5. Use logarithmic differentiation to find y' for y = e^4x
cos(2x) / (x−1)^4 .
6. Show that d/dx (tan (x)) = sec^2 (x) using only your
knowledge of the derivatives of sine/cosine with derivative
rules.
7. Use implicit differentiation to show that...

Find the slope of the line tangent to the curve y=x^2 at the
point (-0.9,0.81) and then find the corresponding equation of the
tangent line.
Find the slope of the line tangent to the curve y=x^2 at the
point (6/7, 36,49) and then find the corresponding equation to the
tangent line.
answer must be simplified fraction

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