Math 221 - Midterm 1 Practice exam 1 - Derivative, Limits, Asymptotes, and Continuity

Sep 30, 2026

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Practice exam for Math 221 Midterm 1 covering core calculus skills. Tasks include using the definition of the derivative as a limit to compute derivatives and determine a tangent line equation, differentiating functions via differentiation rules, and evaluating limits or explaining nonexistence with correct sign/behavior. The exam also requires selecting correct answers from multiple-choice questions, finding horizontal, vertical, and slanted asymptotes, and analyzing a piecewise-defined function for continuity and differentiability at x=1 using parameter k.

LanguageEnglish

CategoryExam

FormatPDF

Pages7

Reading Time18 min

AccessFree to download

Derivatives and Tangent Line

Derivatives from the definition

Tangent line to a given curve

Differentiation Rules

Derivatives of polynomial/trigonometric forms

Limits and Existence

Compute or show limits do not exist

Multiple-Choice Concept Checks

Limits selection and function properties

Asymptotes

Horizontal, vertical, and slanted asymptotes

Continuity and Differentiability of a Piecewise Function

Parameter k for continuity at x=1

Parameter k for differentiability at x=1

Math 221-Midterm 1 Practice exam 1 1. (13 points) (a) (6 points) Use the de􀀌nition of the derivative as a limit to 􀀌nd the derivative of 1 f (x) = x + at x = 1 . x (b) (7 points) Use the de􀀌nition of the derivative as a limit to 􀀌nd the equation of the tangent line to the curve y = px + 1 at x = 3 . 2. (20 points) Calculate the derivatives of the following functions by using di􀀋erentiation rules. (a) y = (10x + 4)2012 (b) y = x sin x cos x (c) y = ~~sin(~~x2~~cos ~~x~~)~~ (d) y = ~~1 ~~n~~c~~x~~os~~x (Please simplify your answer as much as possible) 3. (20 points) Compute the following limits or show that it does not exist. If you think the limit does not exist, explain why, and specify if the limit is 1 or 􀀀1 . jx + 2j (a) lim . x!􀀀 2 x(x + 2) (b) lxim!0 1~~ ~~􀀀~~ ~~coxs(22x) (Hint: Use the double-angle formula cos(2x) = 1 􀀀 2 sin2 (x)) . (c) lim sin(2x) x!1 x (d) lim 1 􀀀 sin x x!􀀙=2 (􀀙 􀀀 2x)2 4. (10 points) For each of the problems, circle the (unique) correct answer. (a) Only one of the following limits exists. Which one? A. lim sin( 1 ) x!1 x B. lim sin x x!1 C. lim sin( 1 ) x!0 x D. lim 1 x!0 sin x (b) xm􀀀1 􀀀 x2~~ ~~􀀀~~ ~~xx~~ ~~􀀀~~ 2~~ is A. 1 B. 􀀀1 C. 0. D. None of the above. (c) Suppose f (x) is an odd function and lim f (x) = 2 . What can we say about lim f (x)? x! 1+ x!􀀀 1+ A. lim f (x) = 2 x!􀀀 1+ B. lim f (x) = 􀀀2 x!􀀀 1+ C. There is not enough information to tell what lim f (x) is. x!􀀀 1+ (d) If f(x) and g(x) are functions de􀀌ned near x = 1 and lim f(x) = 0, then lim f (x) = 0 x! 1 x! 1 g (x) A. True B. False (e) If f0 (0) exists, then lim f(x) = f (0) . x!0 A. True B. False 5. (21 points) Do the following problems about 􀀌nding asymptotes. 2x2 + 2x (a) (7 points) Find all the horizontal asymptotes of f (x) = x2 􀀀 x 􀀀 2. Show your work. (b) (7 points) Find all the vertical asymptotes of f (x) = 2x2x~~ ~~+x~~ ~~x~~2~~. Show your work. (c) (7 points) Find the slanted asymptote of g(x) = 3xx2++~~1~~x as x ! 1 . Show your work. 8 6. (16 points) Consider the function f(x) de􀀌ned by f(x) = <: constant and answer the following questions. x2 k (x 􀀀 1) + 1 if x 􀀔 1 if x > 1 where k is a (a) For what value (or values) of k is f (x) continuous at x = 1? Justify your answer. (b) For what value (or values) of k is f (x) di􀀋erentiable at x = 1? Justify your answer.

FAQs

How do you compute derivatives in this practice exam?

You use the definition of the derivative as a limit for specific problems, then use differentiation rules for other functions.

What types of limit problems are included?

The exam asks to compute limits or determine that a limit does not exist, including using trigonometric identities like the double-angle formula.

How is continuity and differentiability at x=1 handled for the piecewise function?

You determine values of the parameter k that make the piecewise function continuous at x=1, then separately find k values that make it differentiable at x=1.

Sep 30, 2026

Upload by Putri