Difficulty: Intermediate | Prerequisites: Pre-calculus foundations (algebra, geometry, trigonometry). The first study-notes doc in this set.
Limits are the foundation of everything in calculus. The derivative is defined as a limit, and the Fundamental Theorem of Calculus connects derivatives back to integrals through limits. This set of notes covers the core machinery of MTH 132: how limits work, the definition and rules for derivatives, and the three big theorems (IVT, MVT, FToC Part 1) that tie everything together. If you are coming in cold, understand that every topic here builds on the one before it: limits enable derivatives, derivatives enable the Mean Value Theorem, and the Fundamental Theorem closes the loop with integration.
A limit describes what a function approaches as the input approaches a value. A derivative measures instantaneous rate of change and is defined as a limit of a difference quotient. The three key theorems (Intermediate Value, Mean Value, and the Fundamental Theorem of Calculus) guarantee the existence of specific values under continuity and differentiability conditions, and they are tested heavily on exams.
Limit
lim(x→a) f(x) = L means f(x) gets arbitrarily close to L as x approaches a. Think of it as: what value is the function heading towards, even if it never arrives?
One-sided limit
The limit from the left (x→a⁻) or right (x→a⁺) only. A two-sided limit exists if and only if both one-sided limits exist and are equal.
Derivative
f'(x) = lim(h→0) [f(x+h) − f(x)] / h. In simple terms, the derivative is the slope of the tangent line to the curve at a point, or the instantaneous rate of change.
Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and N is between f(a) and f(b), then there exists some c in (a, b) where f(c) = N. Think of it as: a continuous function cannot jump over a value. If it starts below a number and ends above it, it must hit that number somewhere in between.
Mean Value Theorem (MVT)
If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) where f'(c) = [f(b) − f(a)] / (b − a). In simple terms, somewhere on the interval, the instantaneous rate of change equals the average rate of change.
Fundamental Theorem of Calculus, Part 1 (FToC P1)
If F(x) = ∫ from a to x of f(t) dt, then F'(x) = f(x). In simple terms, differentiation undoes integration. The derivative of the accumulation function gives you back the original function.
Newton's Method
An iterative formula x_(n+1) = x_n − f(x_n)/f'(x_n) for approximating roots of equations. Think of it as: start with a guess, use the tangent line to get a better guess, and repeat.
Two-sided limit existence: lim(x→a) f(x) exists if and only if lim(x→a⁻) f(x) = lim(x→a⁺) f(x). Both one-sided limits must exist and agree.
If they disagree, the two-sided limit does not exist (DNE).
A limit can exist at a point even if the function is undefined there.
Special limit (sine): lim(θ→0) sin θ / θ = 1
This is the backbone of deriving the derivative of sin x. You will see it used repeatedly in proofs.
It only works when the argument of sine matches the denominator. If you have sin(3x)/x, you rewrite it as 3 · sin(3x)/(3x) to apply the identity.
Special limit (cosine): lim(θ→0) (1 − cos θ) / θ = 0
Used alongside the sine limit in deriving trig derivatives.
A common trick: multiply top and bottom by (1 + cos θ) to convert this into a form involving sin²θ.
Definition: f'(x) = lim(h→0) [f(x + h) − f(x)] / h
This is the "limit definition of the derivative." Early exams often require you to compute derivatives this way before you learn the shortcut rules.
Trigonometric derivatives (memorise all six):
(sin x)' = cos x
(cos x)' = −sin x
(tan x)' = sec² x
(sec x)' = sec x · tan x
(cot x)' = −csc² x
(csc x)' = −csc x · cot x
Patterns to notice:
The "co-" functions (cos, cot, csc) all pick up a negative sign in their derivatives.
The derivatives of tan and cot involve squared functions (sec², csc²).
The derivatives of sec and csc each involve a product of two trig functions.
Intermediate Value Theorem (IVT)
Hypotheses: f is continuous on [a, b], and N is any value between f(a) and f(b), with f(a) ≠ f(b).
Conclusion: there exists at least one c in (a, b) such that f(c) = N.
Typical exam use: proving a root exists. Show f(a) < 0 and f(b) > 0 (or vice versa), cite IVT, conclude there is a root in (a, b).
The theorem says "at least one c exists." It does not tell you what c is or how to find it.
Mean Value Theorem (MVT)
Hypotheses: f is continuous on [a, b] and differentiable on (a, b).
Conclusion: there exists c in (a, b) such that f'(c) = [f(b) − f(a)] / (b − a).
Geometric interpretation: somewhere on the curve, the tangent line is parallel to the secant line connecting (a, f(a)) and (b, f(b)).
Important: if either hypothesis fails (not continuous, or not differentiable), you cannot apply MVT.
Fundamental Theorem of Calculus, Part 1 (FToC P1)
Setup: F(x) = ∫ from a to x of f(t) dt, where f is continuous.
Conclusion: F'(x) = f(x).
In words: the derivative of an integral (with a variable upper limit) gives back the integrand, evaluated at the upper limit.
When the upper limit is not simply x but a function g(x), apply the chain rule: d/dx [∫ from a to g(x) of f(t) dt] = f(g(x)) · g'(x).
Newton's Method: x_(n+1) = x_n − f(x_n) / f'(x_n)
An iterative root-finding technique. Start with an initial guess x₀, compute x₁, then x₂, and so on.
Each step uses the tangent line at the current guess to find a better approximation.
It can fail if f'(x_n) = 0 (division by zero) or if the initial guess is poor.
Summation of a constant: Σ(i=1 to n) c = cn
Adding the same value c a total of n times.
Sum of first n integers: Σ(i=1 to n) i = n(n+1)/2
Example: Σ(i=1 to 100) i = 100(101)/2 = 5050.
Used in Riemann sum calculations.
Sum of first n squares: Σ(i=1 to n) i² = n(n+1)(2n+1)/6
Also used in Riemann sums when the integrand involves x².
Derivatives measure rates of change in every quantitative field: velocity in physics, marginal cost in economics, reaction rates in chemistry. Newton's Method is used by engineering software to solve equations that have no closed-form solution, from structural analysis to orbital mechanics. The Intermediate Value Theorem underpins numerical root-finding algorithms, and the Mean Value Theorem is used in proofs throughout real analysis and applied mathematics (for instance, proving that a car that travels 100 miles in 2 hours must have been going exactly 50 mph at some instant).
Students often think lim(x→a) f(x) = f(a). That is only true when f is continuous at a. A limit can exist at a point where the function is undefined or has a different value.
Many students confuse the MVT conclusion with the IVT conclusion. The MVT is about derivatives (tangent = secant slope). The IVT is about function values (the function hits every value between f(a) and f(b)).
Students frequently drop the negative sign on the derivatives of cos, cot, and csc. All three "co-" derivatives carry a minus sign.
A common error with Newton's Method is forgetting to evaluate both f(x_n) and f'(x_n) at the same point before dividing. Students sometimes compute f at one value and f' at another.
Students sometimes think the FToC says F(x) = f(x). It does not. It says F'(x) = f(x). The theorem involves a derivative on the left side.
⚠️ The limit definition of the derivative is tested early and often. You must be able to compute f'(x) from the definition, not just from shortcut rules.
⚠️ Know the hypotheses of IVT and MVT precisely. Exam questions often present a scenario where one hypothesis fails, and you must identify that the theorem does not apply.
⚠️ All six trig derivatives are expected from memory. They appear in chain-rule, product-rule, and quotient-rule problems throughout the course.
⚠️ The special limits (sin θ / θ → 1 and (1 − cos θ)/θ → 0) appear in limit-evaluation problems and in deriving trig derivatives from the definition.
⚠️ Newton's Method problems typically ask for one or two iterations. The arithmetic is straightforward but easy to fumble under time pressure.
True or false: lim(x→a) f(x) can exist even if f(a) is undefined. _____
Fill in the blank: (cos x)' = _____.
True or false: The MVT requires f to be differentiable on the closed interval [a, b]. _____
Fill in the blank: lim(θ→0) sin θ / θ = _____.
Fill in the blank: Σ(i=1 to n) i = _____.
Answers: 1. True. 2. −sin x. 3. False (differentiable on the open interval (a, b), continuous on the closed interval [a, b]). 4. 1. 5. n(n+1)/2.
Q: Use the limit definition to find the derivative of f(x) = x².
A: f'(x) = lim(h→0) [(x+h)² − x²] / h = lim(h→0) [2xh + h²] / h = lim(h→0) (2x + h) = 2x.
Q: Evaluate lim(x→0) sin(5x) / x.
A: Rewrite as 5 · sin(5x)/(5x). As x→0, 5x→0, so sin(5x)/(5x) → 1. The answer is 5.
Q: Show that x³ + x − 1 = 0 has a root on (0, 1).
A: f(x) = x³ + x − 1 is continuous everywhere. f(0) = −1 < 0 and f(1) = 1 > 0. By the IVT, there is some c in (0, 1) with f(c) = 0.
Q: A car travels 150 miles in 3 hours. Use the MVT to conclude something about its speed.
A: The average speed is 150/3 = 50 mph. By the MVT, there is some instant during the trip where the car's instantaneous speed was exactly 50 mph.
Q: Let F(x) = ∫ from 2 to x of (t³ + 1) dt. Find F'(x).
A: By FToC Part 1, F'(x) = x³ + 1.
Q: Use one iteration of Newton's Method to approximate a root of f(x) = x² − 3, starting with x₀ = 2.
A: f(2) = 1, f'(x) = 2x, so f'(2) = 4. x₁ = 2 − 1/4 = 1.75.
Limits connect back to the algebraic factoring skills in the pre-calculus notes: most limit problems at the MTH 132 level reduce to factoring, cancelling, and then substituting. Derivatives connect forward to optimisation (finding maxima and minima), related rates, and curve sketching, all of which rely on computing and interpreting first and second derivatives. The FToC connects differentiation to integration, which is the central theme of Calculus 2. The summation formulas connect to Riemann sums, which are the bridge between finite sums and definite integrals.
Limit, one-sided limit, left-hand limit, right-hand limit, two-sided limit, limit does not exist, DNE, squeeze theorem, derivative, differentiation, limit definition of derivative, difference quotient, instantaneous rate of change, slope of tangent line, trig derivatives, sin cos tan sec csc cot derivatives, Intermediate Value Theorem, IVT, root existence, Mean Value Theorem, MVT, average rate of change, Rolle's Theorem, Fundamental Theorem of Calculus, FToC, FTC, antiderivative, accumulation function, Newton's Method, Newton-Raphson, root approximation, iterative method, summation formulas, Riemann sum, sigma notation, MTH 132, Calculus 1, Michigan State University.