(-2) + 6 + 1 = 1 + 6 + (-2)
-2 + 7 = 7 - 2
5 = 5
3/5 = 12/20
2/4 = 10/20
Since 12 is greater than 10, Vinny ate more
Answer:
To isolate variables you can either:
- subtract from both sides
- add to both sides
- multiply both sides (by the same number )
- divide from both sides
These were basic.
Some more advanced ones are :
- raise both sides to the same exponent
- root both sides by the same root.
It has to be an inverse operation however, for all of these properties. And remember, they only work because they are done to both sides.
Explicit Functiony = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation and does not appear at all on the other side. (ex. y = -3x + 5)Implicit FunctionAn equation in which y is not alone on one side. (ex. 3x + y = 5)Implicit DifferentiationGiven a relation of x and y, find dy/dx algebraically.d/dx ln(x)1/xd/dx logb(x) (base b)1/xln(b)d/dx ln(u)1/u × du/dxd/dx logb(u) (base b)1/uln(b) × du/dx(f⁻¹)'(x) = 1/(f'(f⁻¹(x))) iff is a differentiable and one-to-one functiondy/dx = 1/(dx/dy) ify = is a differentiable and one-to-one functiond/dx (b∧x)b∧x × ln(b)d/dx e∧xe∧xd/dx (b∧u)b∧u × ln(b) du/dxd/dx (e∧u)e∧u du/dxDerivatives of inverse trig functionsStrategy for Solving Related Rates Problems<span>1. Assign letters to all quantities that vary with time and any others that seem relevant to the problem. Give a definition for each letter.
2. Identify the rates of change that are known and the rate of change that is to be found. Interpret each rate as a derivative.
3. Find an equation that relates the variables whose rates of change were identified in Step 2. To do this, it will often be helpful to draw an appropriately labeled figure that illustrates the relationship.
4. Differentiate both sides of the equation obtained in Step 3 with respect to time to produce a relationship between the known rates of change and the unknown rate of change.
5. After completing Step 4, substitute all known values for the rates of change and the variables, and then solve for the unknown rate of change.</span>Local Linear Approximation formula<span>f(x) ≈ f(x₀) + f'(x₀)(x - x₀)
f(x₀ + ∆x) ≈ f(x₀) + f'(x₀)∆x when ∆x = x - x₀</span>Local Linear Approximation from the Differential Point of View∆y ≈ f'(x)dx = dyError Propagation Variables<span>x₀ is the exact value of the quantity being measured
y₀ = f(x₀) is the exact value of the quantity being computed
x is the measured value of x₀
y = f(x) is the computed value of y</span>L'Hopital's RuleApplying L'Hopital's Rule<span>1. Check that the limit of f(x)/g(x) is an indeterminate form of type 0/0.
2. Differentiate f and g separately.
3. Find the limit of f'(x)/g'(x). If the limit is finite, +∞, or -∞, then it is equal to the limit of f(x)/g(x).</span>
The answer is 0.833333333
