Contextual Applications of Differentiation
Analytical Applications of Differentiation
Differential Equations
Final Jeopardy
100
A particle moves along a horizontal line such that its position is s=2t^3-9t^2+12t-4 for t>0
Find all t for which the particle is moving to the right and which the velocity is increasing
Particle moves to the right when t<1 and t>2
V increases when t>3/2
100
For 4x^3-6x^2-8, what are the critical points?
x=0, x=1
100
Match the correct differential equation to the slope field
A
100
A particle moves along the x-axis for t_>0. The velocity of the particle at time t is given by v(t)= -3+2e^cos(t^2/2). The particle is at position x=3 at time t=3
a) At time t=2, is the particle speeding up or slowing down?
b) Find all times t in the interval 0<t<2 when the particle changes direction. Justify your answer
c) Find the position of the particle at time t=0
d) Find the total distance the particle travels from time t=0 to time t=2
a) v(2)= -0.469
a(2)= -3.279
The particle is speeding up since the velocity and acceleration have the same sign
b) v(t)=0 at t=1.860
The velocity changes from positive to negative at t=1.860 on 0<t<2; therefore, the particle changes direction at t=1.860
c) x(0)=x(2) +integral from 2 to 0 v(t)dt = 3+ (-3.421) = -0.421
d) Distance = integral 0 to 2 absolute value of v(t) dt = 3.487
200
s=2t^3-9t^2+12t-4
Find all t for which the speed of the particle is increasing
Find the speed when t=3/2
The speed of the particle is increasing for t>2, 1<t<3/2
The speed of the particle when t=3/2 is 3/2
200
For what values of x is f(x)=x^4-4x^3 increasing and for what values is it decreasing?
Decreasing from negative infinity to 3, Increasing from 3 to positive infinity.
200
Which slope field is generated by the differential equation dy/dx= x-y
C
300
Find the tangent line approximation for sin(x) at a=0
=x
300
Find any maximum, minimum, or inflection points of the graph f(x)=x^3-5x^2+3x+6
Local maximum: (1/3, 175/27)
Local minimum: (3, -3)
Point of Inflection: x=5/3
300
Solve dy/dx=-x/y given the initial condition of y(0)=2
y=(4-x^2)^1/2
400
Find the tangent line approximation for 2x^3-3x a=1
3x-4
400
A charter bus company advertises a trip for a group as follows: At least 20 people must sign up. The cost when 20 participate is 80$ per person. The price will drop by 2$ per ticket for each member of the traveling group in excess of 20. If the bus can accommodate 28, how many participants will maximize the company's revenue?
28 participants will maximize revenue
400
The population of a country is growing at a rate proporitional to its population. If the growth rate per year is 4% of the current population, how long will it take for the population to double?
17.33 years
500
If one leg AB of a right triangle increases at a rate of 2 inches per second while the other leg AC decreases at a 3 inches per second, how fast is the hypotenuse changing when AB= 6 feet and AC= 8 feet
-1/10 ft/sec
500
The volume of a cylinder equals V cubic inches, where V is a constant. Find the proportions of the cylinder that minimize the toal surface area.
The total surface area of a cylinder of fixed volume is a minimum when its height is equal to its diameter.
500
At a yearly rate of 5% compounded continuously, how long does it take to the nearest year for an investment to triple?
22 years