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Rate of Change Formulas
Optimization Formulas
True or False
Quick Problems
100
The slope formula is...?
(y₂-y1) / (x₂-x1)
100
What is the formula for rectangular area?
Area = length x width
100
True or false: Speed can be positive or negative.
False. Speed is always positive! Velocity can be positive or negative.
200
Does rate equal distance/time or time/distance?
distance/time
200
What is the formula for the perimeter of a rectangle?
Perimeter of rectangle = 2 x length + 2 x width
200
True or false: Instantaneous velocity is not the derivative of the position function.
False. Instantaneous velocity *is* the derivative of the position function.
300
f(b)-f(a) / (b-a) finds the...?
The average velocity when given intervals.
300
√(x²+y²) is the formula for the...?
Hypotenuse
300
True or false: Acceleration is the second derivative of the position function.
True. Acceleration is the second derivative of the position function.
300
If s(t)= 3t+1 is a measure of feet traveled per second, find the average velocity between t=0 and t=3.
s(0)=1 s(3)=10 f(b)-f(a)/b-a 10-1/3-0 = 9/3 = 3ft/sec
400
The derivative of a position function is called the ______ function.
The derivative of a position function is called the *velocity* function.
400
What is the formula for the volume of a rectangular solid?
Volume of rectangular solid = length x width x height
400
True or false: In optimization, if there is more than one variable in the primary equation, you must write a secondary equation.
True. If there is more than one variable in the primary equation, you must write a secondary equation.
400
Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.
x+2y=100 x=100-2y P=xy P=(100-2y)y P=100y-2y^2 P'=100-4y=0 x=100-2y y=25 x=50
500
What is the velocity function equation?
V(t)= (lim(Δx->0)) (s(t+Δt)-s(t)) / (Δt) = s'(t)
500
What is the formula for the volume of a cylinder?
Volume of cylinder = π(radius²) x height
500
True or false: When you do an optimization problem, the critical point of the function will always tell you what the absolute maximum or absolute minimum is.
False. The critical point of the function will give a relative max/min.
500
A Florida Citrus grower estimates that if 60 orange trees are planted; the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. How many trees should the grower plant to maximize the total yield?
Let n= the number of additional trees. Let Y= the total yield = number of trees × the yield per tree. Then: Y (n) = (60trees + n · trees)(400oranges − n · 4oranges) = (60 + n)(400 − 4n) = 24, 000 + 160n − 4n^2 to maximize! Lets find the critical numbers: Y'(n) = 160 − 8n = 0 ⇒ n =160/8 = 20 is the only critical number. Moreover, Y"(n) = −8 ⇒ R"(20) = −8 < 0. By the second derivative test, Y has a local maximum at n = 20, which is an absolute maximum since it is the only critical number. The grower should plant 60+20 = 80 trees to maximize the total yield.