Tutorfly Review Session Math 31A
- 1. MATH 31A Review Ege Tanboga Tutorfly
- 2. Continuity Keep in mind that continuity at a point x = c requires that: 1. f (c) is defined. 2. lim x->c f (x) exists. 3. They are equal. Means limit from both sides should be equal
- 3. Squeeze Theorem If a function has certain boundaries it cannot go above / below, then you can squeeze it in between those boundaries to find a limit Ex. xsin(1/x)
- 4. Intermediate Value Theorem A function cannot skip values if it is continuous If the boundaries change sign (Ex. f(a) < 0 f(b) >0), then there is a point where f(c)= 0, between a and b). This is a consequence of IVT
- 5. Differentiability Differentiability implies continuity: If f is differentiable at x = a, then f is continuous at x = a. However, there exist continuous functions that are not differentiable. Ex. Absolute value function Chain Rule: dy/dx = (dy/du) (du/dx) This logic is used in both implicit differentiation and related rates so it is important to understand this notation of Chain Rule
- 6. Related Rates Goal: calculate an unknown rate of change in terms of other rates of change that are known. 3 Steps 1. Identify what you have and what you need to find 2. Find an equation to relate variables (relate known to unknown) – don’t substitute until you computed all derivatives 3. Use given data to find unknown derivative
- 7. Extreme Values / Applications of Derivative Critical points when the derivative is 0 or undefined Plugging in values between those critical points, we can learn about the behavior of our function (sometimes without needing the second derivative test) We can use the second derivative test to improve out understanding even more Concavity -> can be found using either second derivative test or through plugging values between critical points for the first derivative Inflection Point: If f ′′(c) = 0 or f ′′(c) does not exist and f′′(x) changes sign at x = c, then f has a point of inflection at x = c.
- 8. First Derivative f′ >0 ⇒ f is increasing f′ <0 ⇒ f is decreasing Second Derivative f′′ >0 ⇒ f is concave up (local minimum) f′′ <0 ⇒ f is concave down (local maximum) If second derivative test is inconclusive, go back to first derivative test
- 9. Optimization Choose which variables are relevant Find the function and the interval you are interested in (if the function has more one variable, use the constraints to reduce it to only one variable) Optimize the function using knowledge about extreme values If bounds are not included, f may not take a min or max! You need to check limit near bounds and the values at critical points
- 10. Integrals – Fundamental Theorem of Calculus FTC 1 – we can use antiderivatives to calculate definite integrals -> integration as we know it with bounds FTC 2 – more important Conditions: f is continuous on an open interval I and a is a point in I 𝑑 𝑑𝑥 𝑎 𝑥 𝑓 𝑡 𝑑𝑡 = 𝑓(𝑥) If the bounds are both related to x, factor out the integral and evaluate separately If bounds are not just x (Ex. x^3), take the derivative of the bounds as well (3x^2) - > comes from chain rule
- 11. Integrals - substitution Try to see the derivative of a function inside an integral If you change the variable, you need to change the bounds to your new variable as well Might need to do substitution twice in some cases
- 12. Areas and Volumes Area: Integral of top-bottom Volumes of Revolution: If rotated around x-axis, then the general formula below can be applied to any situation 𝑎 𝑏 𝑓 𝑥 2 − 𝑔 𝑥 2 𝑑𝑥 see how similar it is to the area formula If it is rotated around the y axis, find functions in terms of y, and treat y as if it was x (think of it as we are naming the variable with a different letter)
- 13. General Final Tips Pay attention to concepts Make sure you understand the question before proceeding Don’t rush, finals are not as time intensive as midterms (in general) In related rates / optimization questions, always write out what you have and what you want Don’t panic if your second derivative test is inconclusive, you can always go back to first derivative test to figure things out If asked for a derivative of an integral, use FTC (otherwise you will end up trying to solve a complex integral / waste time)