Calculus Answers

Triangle ABC has vertices A (4, 7, 7), B (1, 6, 5) and C (-2, 9, 8). What kind of triangle is ΔABC? Justify your answer.

A vector has direction angles α = 85° and β = 65°
a) Find the value of γ?
b) Find a vector that has those direction angles?
c) Explain why it is not possible for two of a vector's direction angles to be less than 45°?

Sketch the graph of the function y = sin(x^2) for -2π ≤ x ≤ 2π
x and y-intercepts.?
Sign chart.?
Horizontal, vertical and slant asymptotes.?
End behaviour.?
Critical points.?
Slope chart.?
Points of inflection.?
Concavity chart.?

Sketch the graph of the function y = sin(x^2) for -2π ≤ x ≤ 2π.

Find the function’s domain and range. Next, describe level curves of the funtions
a)f(x,y)=x-y
b)f(x,y)=x^2+y^2
c)f(x,y,z)=x+y+z

Minimize the function f subject to two constraints:f(x,y,z)=xyz on the intersection of x^2+y^2-1=0 and x-z=0
Here we have two constraints g1=0 and g2=0 and thus need two multipliers λ and μ and the system becomes ∇f =λ∇g1+μ∇g2, g1=0 , g2=0.

The population (P) of an island y years after colonisation is given by the function
P =250/1+4e^(-0.01y)
What was the initial population of the island?
How long did it take before the island had a population of 150?
After how many years was the population growing the fastest?
Sketch the function.
Give a possible explanation for the shape of the curve.

Sketch the graph of the function y = sin(x^2) for -2π ≤ x ≤ 2π

Give an example of a situation in which composite differentiation might be used. Give examples of functions that might be applicable in your situation, and show how the relevant rates of change might be calculated.

Minimize the function f subject to two constraints:
f(x,y,z)=xyzon the intersection of x^2+y^2-1=0 and x-z=0
Here we have two constraints g1=0 and g2=0 and thus need two multipliers λ and μ and the system becomes ∇f=λ∇g1+μ∇g2 , g1=0, g2=0