Calculus Answers

Parameterize f(x) = sin(2x) from x=−π to x=π.

Find parametric equations for the tangent line to the curve r(t) =e^(−t)〈cos(t),sin(t),1〉at t= 0.

The cost (in dollars) of producing x units of a certain commodity is C(x)=500+10x+0.05x^2

I) find the average rate of change of C with respect x when the production level is from x=100 to x=105 units
ii) find the instantaneous rate of change of C with respect to x=100

Obtain the Taylor series for f(x) = √x at a=4

State whether the following statements are true or false. Give reasons for your answers.
a. the function f:R³→R, givan by f(x,y,z)=|x|+|y|+|z| is differential at (2,3,-1)

Suppose you are driving a van down a highway. Use dimensional analysis to find the wind force
you are experiencing, assuming that the force is affected by the wind density, the speed of the van
and its surface area exposed to the wind direction.

For r(t)=⟨tsin(t),e^t,t+π⟩, then r′(t) is equal to
Select one:
a. ⟨sin(t)−tcos(t),e^t,1⟩+c where c is an arbitrary constant vector

b. ⟨sin(t)+tcos(t),e^t,t⟩


c. ⟨cos(t),e^t,π⟩


d. ⟨sin(t)−tcos(t),e^t⟩


e. ⟨sin(t)+tcos(t),e^t,1⟩

For r(t)=⟨tsin(t),e^t,t+π⟩, then r′(t) is equal to
Select one:
a. ⟨sin(t)−tcos(t),e^t,1⟩+c where c is an arbitrary constant vector

b. ⟨sin(t)+tcos(t),e^t,t⟩


c. ⟨cos(t),e^t,π⟩


d. ⟨sin(t)−tcos(t),e^t⟩


e. ⟨sin(t)+tcos(t),e^t,1⟩

For r(t)=⟨tsin(t),e^t,t+π⟩, then r′(t) is equal to
Select one:
a. ⟨sin(t)−tcos(t),e^t,1⟩+c where c is an arbitrary constant vector

b. ⟨sin(t)+tcos(t),e^t,t⟩


c. ⟨cos(t),e^t,π⟩


d. ⟨sin(t)−tcos(t),e^t⟩


e. ⟨sin(t)+tcos(t),e^t,1⟩

The limt→∞r(t) where r(t)=⟨e^(−2t),4arctan(t),−4⟩ is
Select one:
a. 4
b. ⟨0,2π,−4⟩

c. undefined
d. π+4


e. ⟨0,0,4⟩