# Northwestern MATH 220-1 Differential Calculus: Topics and Practice > A study guide for Northwestern MATH 220-1 Single-Variable Differential Calculus: limits, derivatives, related rates, the MVT, curve sketching and optimization. Source: https://practicecalculus.com/courses/northwestern-math-220-1 Updated: 2026-11-08 ## At a glance - **Course code:** MATH 220-1 - **Units:** 1 unit - **Offered:** Fall and Winter quarters only - **Prerequisites:** None listed in the course description; AP Calculus AB scores of 1 to 3 are placed here - **Covers:** Limits, differentiation, linear approximation and related rates, the Extreme Value Theorem, the Mean Value Theorem, curve sketching and optimization - **Next course:** MATH 220-2 (Single-Variable Integral Calculus) - **Practice here:** [Calculus 1](/learn/calculus-1), units 1 to 5 ## Key takeaways - MATH 220-1 is the first quarter of Northwestern's standard calculus sequence and is all about derivatives. - It covers limits, differentiation, linear approximation, related rates, the Extreme Value and Mean Value Theorems, curve sketching and optimization. - It is offered in the Fall and Winter quarters only. - Integration waits until MATH 220-2, so 220-1 is narrower than a semester-long Calculus 1 course. **Northwestern MATH 220-1**, Single-Variable Differential Calculus, is the first course in Northwestern's standard calculus sequence. Because Northwestern runs on quarters, the course focuses on one big idea, the derivative, and leaves integration to MATH 220-2. This guide lists what the course covers and links each topic to free practice with instant feedback. ## What MATH 220-1 covers Northwestern's course description lists: limits; differentiation; linear approximation and related rates; the Extreme Value Theorem, the Mean Value Theorem and curve sketching; and optimization. The course is worth one unit. The math department's First-Year Focus page lists MATH 220-1 as part of the standard calculus and linear algebra courses and says it is offered in the Fall and Winter quarters only. The course description does not list a prerequisite, but it notes that you cannot earn credit for both MATH 220-1 and courses such as MATH 211-0 or MATH 218-1 and 218-2. Compared with a semester-long **Calculus 1** course, MATH 220-1 covers the derivative half in depth and stops before antiderivatives and the definite integral. ## MATH 220-1 topic map | MATH 220-1 topic | Practice | Read | | --- | --- | --- | | Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits) | | Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity guide](/blog/continuity) | | The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) | | Differentiation rules | [Sum and constant rules](/learn/calculus-1/derivative-rules/sum-and-constant-rules) | [Power rule](/rules/power-rule), [quotient rule](/rules/quotient-rule) | | Chain rule | [Chain rule with powers](/learn/calculus-1/derivative-rules/chain-rule-powers) | [Chain rule guide](/blog/chain-rule) | | Implicit differentiation | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Implicit differentiation guide](/blog/implicit-differentiation) | | Linear approximation | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) | | Related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates guide](/blog/related-rates) | | Extreme Value Theorem | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem) | | Mean Value Theorem | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem) | | Curve sketching | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) | | Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) | ## Placement: 220-1, 218-1 or a later course? The department recommends MATH 220-1 for students with AP Calculus AB scores of 1 to 3, while a 4 places into MATH 220-2. For students with little or no calculus, or who are not confident in their preparation, the department offers MATH 218-1,2,3, Single-Variable Calculus with Precalculus. It covers the same calculus as 220-1,2 plus precalculus review across three quarters, and the department stresses that it is not an easier sequence. ## Example problems at the MATH 220-1 level **1. A related rate.** A 10-foot ladder slides down a wall. When the base is 6 feet out and moving away at 1 ft/s, $x\frac{dx}{dt} + y\frac{dy}{dt} = 0$ with $y = 8$ gives $\frac{dy}{dt} = -\frac{6}{8} = -0.75$ ft/s. **2. The Mean Value Theorem.** For $f(x) = x^2$ on $[1, 3]$, the average rate is $\frac{9 - 1}{2} = 4$, and $f'(c) = 2c = 4$ at $c = 2$. **3. Linear approximation.** $\sqrt{4.1} \approx 2 + \frac{1}{4}(0.1) = 2.025$. ## How to do well in MATH 220-1 - **Keep pace with the quarter.** Ten weeks go quickly, so practice a little every day rather than catching up before midterms. - **Get the derivative rules automatic early.** Related rates and optimization are much easier once differentiation takes no effort. - **Practice word problems.** Related rates and optimization reward a routine: draw, label, write the equation, differentiate. - **Know the theorems' conditions.** The Extreme Value and Mean Value Theorems have hypotheses, and exam questions often test them. Next comes integration: see the [MATH 220-2 guide](/courses/northwestern-math-220-2). ## Frequently asked questions ### What is Northwestern MATH 220-1? MATH 220-1, Single-Variable Differential Calculus, is a one-unit course covering limits, differentiation, linear approximation and related rates, the Extreme Value Theorem, the Mean Value Theorem, curve sketching and optimization. ### When is MATH 220-1 offered? The math department's First-Year Focus page says MATH 220-1 is offered in Fall and Winter only. ### Should I take MATH 220-1 or MATH 218-1? MATH 218-1,2,3 covers the same calculus as 220-1,2 plus precalculus review over three quarters. The department says 218 is not easier, and is meant for students with little or no calculus or who are not confident in their preparation. ### Which AP scores place into MATH 220-1? The department recommends MATH 220-1 for AP Calculus AB scores of 1 to 3. A 4 places into MATH 220-2. ### Does MATH 220-1 cover integrals? No. Definite integrals, antiderivatives and the Fundamental Theorem of Calculus are in MATH 220-2.