This document provides a review of key calculus concepts for MATH 31A including continuity, the squeeze theorem, the intermediate value theorem, differentiability, chain rule, related rates, optimization, the fundamental theorem of calculus, integration by substitution, areas, volumes, and general tips for the final exam. Key points covered include the definition of continuity, how to apply the squeeze theorem and intermediate value theorem, the relationship between differentiability and continuity, how to set up and solve related rates problems, how to find critical points and extreme values using derivatives, and how to integrate using the fundamental theorem of calculus and substitution.

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)