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How to implicitly differentiate
Think of “y” as a function, so use chain rule on “y”
Isolate dy/dx, if necessary factor it out
Solve for dy/dx
How to implicitly find a second derivative
Take the first derivative implicitly
Take the derivative again, and sub in the first derivative equation for any dy/dx that appears in the equation
Implicit diff of y²
2y dy/dx
Implicit diff of 2y
2 dy/dx
Implicit diff of sin(x + y)
cos(x + y) * (1 + dy/dx)
Implicit diff of ln(3y⁷)
(21y⁶ dy/dx) / 3y⁷
u’/u
Will ex ever equal 0?
NO
Strategies for related rates
Draw a picture
Make a list of all knowns and unknowns
After plugging in constants, differentiate with respect to t, d/dt
Substitute the known quantities and rates to solve
Can you substitute a non-constant quantity before differentiating?
NO
Plug in constants that NEVER CHANGE before…
differentiating
Take the derivative of variables that could ________ in the problem
change
How to solve cone related rates
Set up similar triangles to write a missing variable in terms of another (if you’re missing r, use the proportion to write r in terms of h)
Set up proportion between dimensions of the ENTIRE cone, and proportion between the variable dimensions of the water level in the cone
Differentiate after subbing in variables
How to solve related rates inequality problems
If the area is always increasing then dA/dt > 0
Plug in differentiated equation into inequality if you can make one based on the info given
Only write the positive answer if…
the direction of the answer is given in the problem (can’t be moving downward at a negative rate, double negative)
How to solve angle of elevation related rates
Set up of trig equation and differentiate
Find θ using inverse trig after differentiating because it’s not constant
When differentiating, constant values go to…
0
How to solve walking person’s shadow related rates
Find a proportion between the similar triangles between the person’s height and their shadow
The only constants are the total length of the floor and the person’s height
Differentiate the proportion, DON’T CROSS MULTIPLY
If you’re missing values for a variable in a related rates problem, then…
represent the variable in terms of another that you know the value of
Plug in given numbers for changing values (height or another dimension that has a changing rate)…
AFTER differentiating, because you need the derivative of the value if it’s changing
Optimization strategies
Draw a picture (if applicable) and identify knowns and unknowns
Write an equation (model) that will be optimized
Write your equation in terms of a single variable
Determine the desired max or min value with candidates test or 2nd derivative test
Determine the domain (endpoints) of the equation to verify if the endpoints represent a max/min
How to solve distance between a graph and point optimizations
Use the distance formula and for the 2nd coordinate, keep in terms of x
Differentiate the distance formula
The answer should be a coordinate on the graph where the distance is either the smallest or greatest
How to solve most optimization problems
Create an equation that has all the same variables
In order to make all the variables the same, set up other equations and plug in
Differentiate the entire equation
Set the derivative equal to zero in order to find critical points
Plug critical points into the 2nd derivative test to verify max/mins
How to solve gutter angle optimizations
Use the trapezoid formula
Use opposite interior angles to set the trapezoid angles equal to the angles of elevation on the outside of the object
Find the height and bases of the trapezoid using sinθ
The domain of the angle can’t be greater than π/2 because the gutter will fold in and lose area
For real life optimization problems, don’t forgot to find the…
DOMAIN
the domain can cancel out some critical values that are found from the derivative of the optimization equation, narrowing the answer to 1!!!!
How to solve water and land optimizations (river vs land travel)
Draw a right triangle connecting the two sides, and a line going down from the bottom edge of the right triangle representing land travel
The hypotenuse represents water travel
Total distance traveled is the hypotenuse plus the total length minus the length of the segment connected to the right triangle
If substituting one variable for another BEFORE differentiating doesn’t work because you get a rate that you don’t have or another issue, then try…
substituting the variable AFTER differentiating the original equation
How to solve number puzzle optimizations
Set up two equations, one equal to a numerical constant, and another equal to the constant P
Rewrite the first equation in terms of x, and sub in for y in the other equation
When differentiating the optimized equation, set P equal to 0 because it’s just some constant
To verify using candidate’s test, plug calculated x values back into the equation with x and P, and look for max/mins
Don’t forget to add _____ to answers for related rates and optimization problems
UNITS!!!!!
For optimization problems, differentiate with respect to…
x (d/dx)
For related rates problems, differentiate with respect to…
t (d/dt)
Equation setup for finding the most economical path over land and water
Total cost = (land cost)(distance on land) + (underwater cost)(distance in water)
Equation setup for finding when someone should switch to land travel to arrive in the shortest time
time = distance/rate
How to find a vertical tangent using implicit differentiation
Set the denominator of the derivative equal to zero and create a function in terms of x
sub in the equation for y back into the og equation
Always include ________ in calculations
negatives
