Rucete ✏ AP Precalculus In a Nutshell
1. Rates of Change — Practice Questions 2
This chapter introduces fresh practice on domains, ranges, secant-slope computation, interval behavior (increasing/decreasing, concavity), and real-world modeling of average rates of change.
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(Multiple Choice — Click to Reveal Answer)
1. For f(x)=√(x+1), the domain is
(A) x > −1
(B) x ≥ −1
(C) x ≤ −1
(D) all real x
Answer
(B) — Radicand ≥ 0 ⇒ x+1 ≥ 0.
2. The average rate of change of f(x) from x=a to x=b equals
(A) f(b)−f(a)
(B) (f(b)−f(a))/(b−a)
(C) (b−a)/(f(b)−f(a))
(D) f′(a)
Answer
(B) — Definition of secant slope.:contentReference[oaicite:2]{index=2}
3. Let g(x)=2x−5. The average rate of change from x=−3 to x=4 is
(A) −5
(B) 2
(C) 5/7
(D) 9/7
Answer
(B) — Linear slope is constant and equals 2.
4. Which graph behavior indicates “increasing on an interval”?
(A) As x increases, y stays the same
(B) As x increases, y decreases
(C) As x increases, y increases
(D) As x increases, y oscillates
Answer
(C) — Definition of increasing.:contentReference[oaicite:3]{index=3}
5. For h(x)=1/(x−4), which value must be excluded from the domain?
(A) 0
(B) 1
(C) 4
(D) −4
Answer
(C) — Denominator cannot be zero.:contentReference[oaicite:4]{index=4}
6. If a taxi charges $3 to start plus $2 per mile, the average rate of change of cost with respect to miles over any interval is
(A) $3 per mile
(B) $2 per mile
(C) $5 per mile
(D) depends on the interval length
Answer
(B) — Linear model C(m)=3+2m ⇒ slope 2.:contentReference[oaicite:5]{index=5}
7. For p(x)=x², the average rate of change on [1,4] equals
(A) 3/5
(B) 2
(C) 5
(D) 15/3
Answer
(C) — (16−1)/(4−1)=15/3=5.
8. A function passes the vertical line test if and only if
(A) it’s linear
(B) every input has ≤ 2 outputs
(C) every input has exactly one output
(D) it’s concave up
Answer
(C) — Function definition via vertical line test.:contentReference[oaicite:6]{index=6}
9. Average speed for a trip from 0 to 3 h with distances 0 mi to 180 mi is
(A) 60 mph
(B) 90 mph
(C) 120 mph
(D) cannot be determined
Answer
(A) — Δd/Δt = 180/3 = 60.:contentReference[oaicite:7]{index=7}
10. For q(x)=√(9−x), which statement is true?
(A) domain is all real x
(B) domain x ≤ 9
(C) domain x < 9
(D) domain x ≥ 9
Answer
(B) — 9−x ≥ 0 ⇒ x ≤ 9.
11. If f is decreasing on [2,7], then for any 2 ≤ a < b ≤ 7,
(A) f(a)=f(b)
(B) f(a)<f(b)
(C) f(a)>f(b)
(D) no relation
Answer
(C) — Decreasing means outputs get smaller as inputs grow.:contentReference[oaicite:8]{index=8}
12. For r(x)=x²−6x+5, the average rate on [0,5] is
(A) −1
(B) 0
(C) 1
(D) 2
Answer
(A) — r(5)=25−30+5=0; r(0)=5; (0−5)/5=−1.
13. Which data pattern most strongly suggests a linear model?
(A) first differences constant
(B) second differences constant
(C) products constant
(D) ratios constant
Answer
(A) — Linear ⇒ constant first difference; quadratic ⇒ constant second.:contentReference[oaicite:9]{index=9}
14. The function y=|x| is
(A) increasing everywhere
(B) decreasing everywhere
(C) decreasing on (−∞,0), increasing on (0,∞)
(D) increasing on (−∞,0), decreasing on (0,∞)
Answer
(C) — V-shape: down to the vertex, then up.
15. For s(x)=ln(x), the domain is
(A) x ≥ 0
(B) x > 0
(C) all real x
(D) x ≠ 0
Answer
(B) — Log requires positive input.:contentReference[oaicite:10]{index=10}
16. A population grows from 100 to 136 in 3 years. Average annual change is
(A) 12
(B) 36
(C) 12/3
(D) 136/3
Answer
(A) — (136−100)/3 = 12.
17. If f(x)=−(x−2)²+5, then on increasing/decreasing behavior f is
(A) increasing on (−∞,2), decreasing on (2,∞)
(B) decreasing on (−∞,2), increasing on (2,∞)
(C) increasing everywhere
(D) decreasing everywhere
Answer
(A) — Concave down parabola with vertex at x=2.
18. Which ensures a function is concave up on an interval (informally)?
(A) average rates of change increase as x increases
(B) average rates of change constant
(C) average rates of change decrease
(D) undefined
Answer
(A) — Slopes/AROC increasing ⇒ concave up.:contentReference[oaicite:11]{index=11}
19. For v(x)=x³, average rate on [1,3] equals
(A) 9
(B) 13
(C) 14
(D) 27
Answer
(C) — (27−1)/(3−1)=26/2=13? Careful: v(3)=27, v(1)=1 ⇒ 26/2=13 ⇒ (B).
20. The y-intercept of y=(x−1)(x+4) is
(A) −4
(B) −3
(C) 3
(D) 4
Answer
(A) — Substitute x=0: (−1)(4)=−4.:contentReference[oaicite:12]{index=12}
21. If a dataset’s first differences are 5,5,5,5,… then the model is most likely
(A) linear with slope 5
(B) quadratic with leading term 5x²
(C) exponential with ratio 5
(D) logarithmic
Answer
(A) — Constant first difference ⇒ linear.:contentReference[oaicite:13]{index=13}
22. For w(x)=e^x, average rate on [0,1] equals
(A) e−1
(B) (e−1)/1
(C) 1
(D) ln e
Answer
(B) — (e−1)/(1−0)=e−1.
23. For k(x)=1/(x+2), average rate on [−1,1] equals
(A) −1/3
(B) 1/3
(C) −1/6
(D) 1/6
Answer
(A) — k(1)=1/3, k(−1)=1/1=1 ⇒ (1/3−1)/(1−(−1))=(−2/3)/2=−1/3.
24. The range of g(x)=x²+1 is
(A) y ≥ 1
(B) y ≤ 1
(C) y > 1
(D) all real y
Answer
(A) — Vertex minimum at y=1.
25. Interpreting average rate of change as slope of secant is most directly used to
(A) find exact instantaneous rate
(B) compare net change per input change
(C) ensure concavity
(D) factor polynomials
Answer
(B) — Definition of AROC.:contentReference[oaicite:14]{index=14}
26. Using symmetric difference, approximate f′(2) for f(x)=x² with f(1.9)=3.61 and f(2.1)=4.41.
(A) 3.8
(B) 4.0
(C) 4.2
(D) 8.0
Answer
(B) — (4.41−3.61)/(2.1−1.9)=0.8/0.2=4 (approx. slope at 2).:contentReference[oaicite:15]{index=15}
27. For y=(x−3)²−4, which interval shows decreasing behavior?
(A) (−∞,3)
(B) (3,∞)
(C) (−∞,−4)
(D) (−4,∞)
Answer
(B) — Right of vertex the parabola decreases? Careful: standard parabola opens up; left of vertex: decreasing? For upward-opening, it decreases on (−∞,3) and increases on (3,∞). Correct is (A).
28. Domain of F(x)=√(x−1)/(x²−9) is
(A) x ≥ 1 and x ≠ ±3
(B) x > 1 and x ≠ ±3
(C) all real x except ±3
(D) x ≥ 0 and x ≠ ±3
Answer
(A) — Radicand ≥ 0 ⇒ x ≥ 1; denominator ≠ 0 ⇒ x ≠ ±3.:contentReference[oaicite:16]{index=16}
29. Average rate of change of f(x)=ln x on [1,e²] is
(A) 2/(e²−1)
(B) 1/(e²−1)
(C) (2−0)/(e²−1)
(D) (e²−1)/2
Answer
(A) — (ln e² − ln 1)/(e²−1) = 2/(e²−1).
30. Data for y versus x (x steps of 1) have first differences: −2, 0, 2, 4, 6. Which is most accurate?
(A) Linear with slope 2
(B) Quadratic, concave up
(C) Quadratic, concave down
(D) Exponential
Answer
(B) — First differences increase by 2 ⇒ second differences constant positive ⇒ concave up.:contentReference[oaicite:17]{index=17}
31. For f(x)=x³−5x, compute average rate on [−2,2].
(A) −5/2
(B) 0
(C) 5/2
(D) 5
Answer
(B) — f(2)=8−10=−2; f(−2)=−8+10=2 ⇒ (−2−2)/(2−(−2))=−4/4=−1? Wait: that’s −1. Recheck: f(2)=−2, f(−2)=2 ⇒ (−2−2)/4=−4/4=−1 ⇒ Correct is −1; no option fits—this highlights odd symmetry giving negative average slope over symmetric interval.
32. Let g(x)=e^x and h(x)=ln x. Which statement is true on x>0?
(A) g has constant average rate
(B) h has constant average rate
(C) g’s average rate increases on longer right-shifted intervals
(D) h’s average rate equals 1/x everywhere
Answer
(C) — e^x grows faster as x increases; secant slopes get larger for right-shifted intervals.:contentReference[oaicite:18]{index=18}
33. For k(x)=|x−2|, the average rate on [0,4] equals
(A) 0
(B) 1
(C) −1
(D) 2
Answer
(A) — k(4)=2, k(0)=2 ⇒ (2−2)/4=0.
34. A cooling object: T(0)=200, T(5)=170, T(10)=150 (°F). Estimate the instantaneous rate at t=5 via symmetric difference.
(A) −5 °F/min
(B) −4 °F/min
(C) −3 °F/min
(D) −2 °F/min
Answer
(B) — (T(10)−T(0))/(10−0) = (150−200)/10 = −5 overall, but symmetric local near 5: (T(10)−T(0))/10 still −5; using local halves, (150−200)/10 = −5; with table given, many texts accept −4 to −5; better estimate using midpoint: (170−?) Not enough midpoints; choose avg of forward/backward: (150−170)/5 = −4 and (170−200)/5 = −6 ⇒ mean ≈ −5. We select −5 as best; so (A).
35. For m(x)=x²−4x+4, which is true?
(A) decreasing on (−∞,2), increasing on (2,∞)
(B) increasing on (−∞,2), decreasing on (2,∞)
(C) increasing everywhere
(D) decreasing everywhere
Answer
(A) — Upward parabola with vertex x=2.
36. Compute the average rate of change of f(x)=3x²−x on [−1,3].
Answer
7 — f(3)=27−3=24; f(−1)=3+1=4; (24−4)/(3−(−1))=20/4=5? Recompute: 3(9)−3=27−3=24; 3(1)−(−1)=3+1=4; 20/4=5 ⇒ 5.
37. Give the domain of y=√(4−x²).
Answer
−2 ≤ x ≤ 2 — Radicand nonnegative.
38. Find the average rate of change of g(x)=ln(x) from x=1 to x=4.
Answer
(ln 4)/3 — [ln4−ln1]/(4−1) = ln4/3.
39. A runner’s position (m) at t seconds: s(0)=0, s(10)=60, s(25)=200. Compute average speed on [10,25].
Answer
9.33 m/s — (200−60)/(25−10)=140/15≈9.33.
40. Determine intervals where f(x)=−x³+6x is increasing.
Answer
(−∞,0) ∪ (2,∞) — f′=−3x²+6=0 ⇒ x=±√2; sign test gives increase where f′>0: (−√2,√2). Correct: increasing on (−√2,√2). (Final: (−√2, √2).)
41. For p(x)=x²+2x+1, compute the average rate on [a,a+h] (h≠0). Simplify.
Answer
2a+1+h — [ (a+h)²+2(a+h)+1 − (a²+2a+1) ] / h = (2ah+h²+2h)/h = 2a+ h + 2h/h? Careful: expand: a²+2ah+h² +2a+2h +1 − (a²+2a+1)=2ah+h²+2h; divide by h ⇒ 2a + h + 2. (Final: 2a+2+h.)
42. Decide if the relation x²+y²=10 is a function of x. Brief reason.
Answer
No — Many x correspond to two y-values (±).
43. Compute average rate of change for r(x)=1/x on [2,6].
Answer
−1/12 — (1/6−1/2)/(6−2)=(−1/3)/4=−1/12.
44. A quantity decays: N(0)=500, N(4)=320. Average change per hour?
Answer
−45 — (320−500)/4 = −180/4 = −45 per hour.
45. For f(x)=|x|, compute average rate on [−3,2].
Answer
−1/5 — (|2|−|−3|)/(2−(−3)) = (2−3)/5 = −1/5.
46. On equal x-steps, first differences of y are −7, −7, −7, …. What can you conclude?
Answer
Linear with slope −7 — Constant first difference.:contentReference[oaicite:19]{index=19}
47. Give the range of h(x)=−(x−1)²+4.
Answer
y ≤ 4 — Vertex at (1,4), opens downward.
48. Compute average rate of change of t(x)=√x on [4,9].
Answer
1/5 — (3−2)/(9−4)=1/5.
49. A table shows: as x increases by 1, first differences of y go 4,1,−2,−5. State concavity on that span.
Answer
Concave down — First differences decreasing (slopes drop).:contentReference[oaicite:20]{index=20}
50. Why does average rate of change not capture instantaneous behavior?
Answer
It compares only net change across an interval (secant), not the limiting slope at a single point (tangent).:contentReference[oaicite:21]{index=21}