Limits, continuity, algebra review
Chain Rule, Implicit differentiation
Other: Trig/inverse, etc.
Optimization
Graphing, linear approximation
100
f(x) = x3+5x+3
g(x) = -x+2
Find h(x) = f(g(x))
What's an example of how we use this in calculus?
h(x) = (-x+2)3+5(-x+2)+3
Chain rule!!
100
What is the chain rule formula?
What is the general step by step process for implicit differentiation problems?
f(g(x))' = f'(g(x))*g'(x)
Take the derivative of both sides
Use the chain rule (Multiply by dy/dx for y(x) terms
Move all terms with dy/dx to one side and the rest to the other side
factor out dy/dx and divide to isolate
100
What is the formula for inverse derivatives?
What is the derivative of sin, cos? How can you use ratios to find the derivative of any other trig functions?
(f-1(x))' = 1/f'(f-1(x))
d/dx(sin(x)) = cos(x)
d/dx(cos(x)) = -sin(x)
Using properties such as tan(x) = sin(x)/cos(x)
Using sec(x) = 1/cos(x) and csc(x) = 1/sin(x)
Using the inverse formula for arcsin(x), etc.
100
When is it guaranteed that you will find an absolute max/min?
I.e. explain the difference in finding local max/min and absolute.
Absolute max and min can only be guaranteed when the interval is closed and bounded. Ie. [-5,5] because we can test and compare the endpoints. Otherwise, we can find local extrema but we might not know if that is the absolute smallest/biggest value on the graph.
100
What is the general process for graphing a function?
1. Find f'(x)
2. Solve f'(x) = 0
3. Number line - where is f(x) increasing/decreasing
4. Find f''(x)
5. Solve f''(x)
6. Plug in all important points
7. Graph - pay attention to increasing, decreasing, and concavity
8. Determine limits (vertical and horizontal)
200
If k=4, the function is continuous
Solve by setting the piecewise functions equal to each other and plug in 1 for x. Solve for k.
200
Use implicit differentiation to find the second derivative of:
x2 + y2= 25
y'' = (-1/y) -(x2/y3)
Simplifies to -25/y3
200
Find the derivative of
f(x) = cos(x)/4x2
f'(x) = (-xsinx-2cosx)/4x3
200
Find all local extrema for the following function:
f(x) = x3-3x2-9x-1
Local maximum at (-1, 4)
Local minimum at (3, -28)
200
For the following function:
f(x) = x3 - 6x2 + 9x + 30
Determine the intervals where f(x) is concave up/down and what the points of inflection are.
𝑓 is concave down over the interval (−∞,2) and concave up over the interval (2,∞). Since 𝑓 changes concavity at 𝑥=2, the point (2,𝑓(2))=(2,32) is an inflection point.
300
Removable or Nonremovable discontinuity?
f(x) =
-9x if x<= 2
x2 -2x+3 if x>2
Non-removable, jump
300
Find the equation of the line tangent to the graph of y3+x3−3xy=0 at the point (3/2,3/2)
dy/dx = (3y-3x2)/(3y2-3x)
dy/dx at (3/2,3/2) = -1
y=-x+3
300
Find the equation of the line tangent to the graph of 𝑓(𝑥)=𝑥2−4𝑥+6 at 𝑥=1
f'(1) = -2
y-3 = -2(x-1)
300
Find the absolute maximum and absolute minimum of 𝑓(𝑥)=𝑥2−4𝑥+3 over the interval [1,4]
f(2) = -1 is absolute minimum
f(4) = 3 is absolute maximum
300
Where is the function f(x) increasing and decreasing? How about concavity, where is it positive or negative? (Pay close attention to what the graph represents)
Increasing: (-2,-1) U (2,inf)
Decreasing: (-inf, -2) U (0,2)
Concave up: (-inf, -1.5ish) U (1,inf)
Concave down: (-1.5, 1)
400
What are the three criteria for a function to be continuous at a certain point?
1. f(a) exists
2. Lim x--> a of f(x) exists
3. Lim = f(a)
400
Find the derivative of
f(x) = (2x+1)5(3x-2)7
f'(x) = 10(2x+1)4(3x-2)7+21(3x-2)6(2x+1)5
Factor and simplify further
(2x+1)4(3x-2)6(72x+1)
400
Using trig and the inverse formula, find the derivative of sin-1x
1/(1-x2)1/2
400
Owners of a car rental company have determined that if they charge customers 𝑝p dollars per day to rent a car, where 50≤𝑝≤200, the number of cars 𝑛 they rent per day can be modeled by the linear function 𝑛(𝑝)=1000−5𝑝. If they charge $50 per day or less, they will rent all their cars. If they charge $200 per day or more, they will not rent any cars. Assuming the owners plan to charge customers between $50 per day and $200 per day to rent a car, how much should they charge to maximize their revenue?
Revenue = nxp
R(p) = p(1000-5p)
R'(p) = -10p+1000
Interval: [50,200]
R'(p) = 0 at p=100
R(100) is the maximum at $50,000
400
What are the general steps to solving a linear approximation problem?
1. Pick a function (Example if trying to solve square root 16.1, it makes sense to use square root x)
2. Use a base value (Example, 16) and solve for the known point and slope by finding the derivative of the function.
3. Create the equation of the tangent line.
4. Plug in the point we want to approximate.
500
Factor the following function:
f(x) = -4x2 - 21x - 5
f(x) = -(x+5)(4x+1)
500
Find the equation of a line tangent to the graph of
f(x) = 1/(3x-5)2 at x = 2
What are the steps to finding a tangent line equation?
f(2) = 1
h'(x) = -6(3x-5)-3
h'(2) = -6
y-1=-6(x-2)
Find f(x)
Find f'(x)
Plug in to find point and slope
Write in point slope form
500
Where is a function non-differentiable?
Give 3 examples
Corner/cusp (|x|)
Discontinuity/limit does not exists for a point
Vertical tangent
500
A cable television company has its main antenna located at point A on the bank of a straight river 1 Mile wide. It is going to run a cable from point A to point P on the opposite side of the river and then straight along the bank to a town T situated 4 Miles downstream from A. It costs $15,000 per Mile for cable underwater, and $9,000 per mile along the bank. What should the distance from P to T be to minimize the cost.
C = 15,000s + (4-x)9,000
s= (1+x2)1/2
C= 15,000(1+x2)1/2+(4-x)9,000
C' = 15,000*1/2(1+x2)-1/2*2x-9,000
C' = 15,000(1+x2)-1/2-9,000
3(1+x2)1/2=5x
9(1+x2) = 25x2
9 = 16x2
x=3/4
Test endpoints: C(0), C(4)
C(0) = 51,000
C(4) = 61,846.58
C(3/4) = 48,000
Therefore, the distance 4-x should be 4-3/4 = 3.25 to minimize cost.
500
Using linear approximation, find the value of 3√8.1
f(x)~1/12(x-8)+2
f(8.1)~241/120