AP Unit 1 Problems
Thirty problems for AP Unit 1. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.
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Problem 1
A cart has position s(t)=t²+3t meters. At t=2 seconds, find the average rate from 2 to 2+h (h≠0), then the instantaneous rate suggested by its limit.
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Problem 2
Write “the concentration C(t), in milligrams per liter, approaches 8 as t approaches 5 minutes from the left” in limit notation. Explain what this says about C(5).
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Problem 3
Suppose lim(x→4⁻)f(x)=2, lim(x→4⁺)f(x)=2 and f(4)=−1. State lim(x→4)f(x), and explain why it differs from f(4).
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Problem 4
Use the hole graph in Topic 1.3 (text equivalent: f(x)=x+1 for x≠2 and f(2)=5). State the left limit, right limit, two-sided limit and actual value at 2.
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Problem 5
A calculator seems to draw sin(1/x) as a dark band near x=0. Decide whether its limit exists. Explain why the plotting window cannot settle the question.
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Problem 6
| x | 0.9 | 0.99 | 1.01 | 1.1 |
|---|---|---|---|---|
| F(x) | 2.71 | 2.9701 | 3.0301 | 3.31 |
Estimate lim(x→1)F(x). State whether this finite table alone proves the limit.
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Problem 7
Evaluate lim(x→−1)(2x²+3)/(x²+4), stating the condition needed for the quotient limit rule.
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Problem 8
Let f(x)=x² for x<1 and f(x)=2x+1 for x≥1. Find both one-sided limits and decide whether lim(x→1)f(x) exists.
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Problem 9
Evaluate lim(x→−2)(x²+x−2)/(x+2).
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Problem 10
Evaluate lim(x→0)(√(9+x)−3)/x.
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Problem 11
In radians, evaluate lim(x→0)sin(4x)/(3x).
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Problem 12
For each limit, name a suitable first method and then evaluate: (a) lim(x→2)(x²−4)/(x−2); (b) lim(x→0)(√(1+x)−1)/x; (c) lim(x→0⁺)1/x.
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Problem 13
Prove lim(x→0)x²cos(1/x)=0 using the squeeze theorem.
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Problem 14
A function satisfies 5−2|x−1|≤g(x)≤5+3|x−1| for 0<|x−1|<0.2. Find lim(x→1)g(x), and explain whether g(1) must be 5.
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Problem 15
For x≠3, H(x)=(x²−9)/(x−3); H(3) is undefined. Describe its nearby table values at 2.99 and 3.01, the corresponding graph feature, its limit notation, and the verbal meaning.
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Problem 16
Classify the discontinuity at the indicated input: (a) (x²−4)/(x−2) at 2; (b) f(x)=−1 for x<0 and 1 for x≥0 at 0; (c) 1/(x+1)² at −1. Say whether one finite replacement value can repair each.
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Problem 17
Let f(x)=2x+1 for x≠2 and f(2)=4. Test all three continuity conditions at 2.
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Problem 18
Let g(x)=x² when x≤1 and g(x)=2x−1 when x>1. Is g continuous at 1? Give a complete reason.
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Problem 19
Find the maximal intervals of continuity of R(x)=(x+2)/(x²−4). Does cancelling a factor add x=−2 to the original domain?
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Problem 20
On which maximal interval is L(x)=ln(x−2) continuous? Explain the endpoint restriction.
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Problem 21
Define f(x)=(x³−8)/(x−2) for x≠2. Choose f(2) to make f continuous at 2.
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Problem 22
Find a so f(x)=ax+2 for x<1 and f(x)=x²+4 for x≥1 is continuous at 1.
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Problem 23
For V(x)=−2/(x−3), find both one-sided limits at 3 and state the vertical asymptote.
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Problem 24
For W(x)=(x+1)/(x−2)², determine the two-sided infinite-limit behavior at 2. Is +∞ a finite real limit?
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Problem 25
Find both limits at infinity and any horizontal asymptote of F(x)=(4x²−3x+1)/(2x²+7).
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Problem 26
Find lim(x→∞)√(4x²+1)/x and lim(x→−∞)√(4x²+1)/x.
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Problem 27
Use a calculator table at x=2,5,10,20 to inspect x²/eˣ. The established limit as x→∞ is 0. Interpret that limit as a growth comparison, and explain why the four rows alone do not prove it.
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Problem 28
Prove x³−2x−2=0 has at least one solution in (1,2), without finding the solution.
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Problem 29
A continuous tank-temperature function T(t) has T(0)=14°C and T(8)=22°C. Must it reach 19°C during (0,8)? Must it do so exactly once?
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Problem 30
A graph consists of y=−1 for x<0 and y=1 for x≥0. Its endpoint values at −2 and 2 have opposite signs. Explain why IVT cannot establish a zero, and decide whether any zero actually exists.
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More chapter problems
Try these problems before opening the solutions. Show your setup, the hypotheses behind your method, and what your answer means.
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What does a vanishing ratio say?
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Estimate nearby behavior from two displays
Inspect diagram labels
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →