Rucete ✏ AP Precalculus In a Nutshell
1. Rates of Change — Practice Questions
This chapter introduces the concepts of relations, functions, and how to calculate and interpret rates of change.
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(Multiple Choice — Click to Reveal Answer)
1. Which of the following functions has a domain of x > 0?(A) f(x) = x²
(B) f(x) = √x
(C) f(x) = log x
(D) f(x) = e^x
Answer
C — The logarithmic function is only defined for x > 0.2. Which of the following equations represent functions?
I. y − x = 17 + 2x
II. x² + y² = 64
III. x − y² = 81
IV. x² + y = 4
(A) I only
(B) I and III only
(C) I and IV only
(D) I, II, III, and IV
Answer
C — Only I and IV pass the vertical line test; II and III fail because y² leads to multiple outputs for one input.3. Find the average rate of change of f(x) = 3x² − 2 from x = 2 to x = 6.
(A) 24
(B) 18
(C) 12
(D) 10
Answer
A — f(2)=10, f(6)=106; slope = (106−10)/(6−2)=96/4=24.4. Which of the following linear functions is increasing?
(A) s(t) = 4 − t
(B) h(z) = 10
(C) g(x) = −3x + 2
(D) f(x) = −7 + 0.9x
Answer
D — Only the slope 0.9 is positive.5. If a is the zero of f(x) = (x − 3)², evaluate g(a) if g(t) = 10 − 2(t − 1)³.
(A) −6
(B) −3
(C) 1
(D) 3
Answer
A — Zero is a=3. Substituting: g(3) = 10 − 2(2³) = 10 − 16 = −6.6. A pot of boiling water cools as shown in a data table. Approximate the rate of change at t = 3.5 minutes.
(A) −5.04 °F/min
(B) −4.8 °F/min
(C) −4.4 °F/min
(D) −4.14 °F/min
Answer
B — Using the closest interval average slope gives −4.8 °F/min.7. From the given table of values, how does the function behave in 1 ≤ x ≤ 6?
(A) Increasing, concave down
(B) Increasing, concave up
(C) Decreasing, concave down
(D) Decreasing, concave up
Answer
C — The y-values decrease and the second differences are negative, so it is decreasing and concave down.8. Cailey’s Cookies has fixed costs $120/day and variable cost $0.99 per cookie. Which model fits?
(A) C(m) = 0.99m + 120
(B) C(m) = 0.99x + 120
(C) C(m) = 120m + 0.99
(D) C(m) = 0.04125m + 5
Answer
A — Fixed cost plus variable cost per cookie gives 0.99m + 120.9. For f(x) = x² − 4x + 1, evaluate f(x + h).
(A) x² − 4x + 1 + h
(B) h² − 4h + 1
(C) x² + h² + 2xh − 4x − 4h + 1
(D) x² + h² − 4x − 4h + 1
Answer
C — Substitute x+h into the function and expand.10. Which of the following has the domain (−∞, ∞), x ≠ −9, x ≠ 4?
(A) f(x) = √(x−36)
(B) i(x) = (x² + 5x − 36)/(...)
(C) g(x) = 1/[(x+4)(x−9)]
(D) h(x) = x² + 5x − 36
Answer
B — Denominator restriction leads to x ≠ −9 and x ≠ 4.11. The average rate of change of f(x) from x = a to x = b equals
(A) f(b) − f(a)
(B) (b − a)/(f(b) − f(a))
(C) (f(b) − f(a))/(b − a)
(D) f′(c) for some c between a and b
Answer
(C) — By definition, average rate of change = (Δy)/(Δx) = (f(b)−f(a))/(b−a).
12. If a function is increasing on an interval, then for a < b in the interval,
(A) f(a) = f(b)
(B) f(a) > f(b)
(C) f(a) < f(b)
(D) f(a) · f(b) < 0
Answer
(C) — Increasing means larger inputs yield larger outputs.
13. Which statement about domain is true?
(A) All square roots allow negative radicands.
(B) Denominators may be zero if the numerator is also zero.
(C) Logarithms require positive arguments.
(D) Every polynomial excludes at least one x-value.
Answer
(C) — For real logs, the argument must be > 0. Polynomials have domain (−∞, ∞).
14. For the linear function y = −4x + 7, the average rate of change over any interval is
(A) −4
(B) −2
(C) 0
(D) Depends on the interval
Answer
(A) — Linear functions have constant rate of change equal to the slope.
15. A function’s graph passes the vertical line test. Which must be true?
(A) It is linear.
(B) Each input maps to exactly one output.
(C) It has a constant rate of change.
(D) It is differentiable everywhere.
Answer
(B) — The vertical line test characterizes functions (unique output per input).
16. The y-intercept of y = (x − 3)(x + 2)(x + 5) is
(A) −30
(B) −5
(C) 0
(D) 30
Answer
(D) — Set x=0: (−3)(2)(5)= −30? Careful: (0−3)(0+2)(0+5)= (−3)(2)(5)= −30; that is the y-value. Check options: correct is −30. (If (D) was 30, then the correct choice is (A) −30.)
17. The x-intercepts (zeros) of y = 2x − 48 are
(A) x = −48
(B) x = 24
(C) x = 48
(D) None
Answer
(B) — Set 2x − 48 = 0 ⇒ x = 24.
18. A graph is concave up on an interval if, as x increases,
(A) average rates of change are increasing
(B) average rates of change are constant
(C) average rates of change are decreasing
(D) the function is decreasing
Answer
(A) — Concave up corresponds to increasing slopes/average rates.
19. For f(x) = x², the average rate of change from x = 4 to x = 4 + h (h ≠ 0) is
(A) 8
(B) 8 + h
(C) 8 − h
(D) 2h
Answer
(B) — [f(4+h)−f(4)]/h = [(16+8h+h²)−16]/h = 8 + h.
20. If a function’s average rates of change over equal x-steps form a constant sequence, the function is
(A) quadratic
(B) linear
(C) exponential
(D) logarithmic
Answer
(B) — Only linear functions have constant average rate of change.
21. The domain of f(x) = √(5 − 2x) is
(A) x > 0
(B) x ≤ 2.5
(C) x ≥ 2.5
(D) all real numbers
Answer
(B) — Inside radical ≥ 0 ⇒ 5 − 2x ≥ 0 ⇒ x ≤ 2.5.
22. For g(x) = (x − 1)(x + 3), the average rate of change from x = 1 to x = 5 equals
(A) 4
(B) 8
(C) 12
(D) 16
Answer
(C) — g(5)= (4)(8)=32; g(1)=0 ⇒ (32−0)/(5−1)=32/4=8? Correction: (x−1)(x+3)= x²+2x−3; g(5)=25+10−3=32, g(1)=0 ⇒ 32/4= 8. So the correct choice is (B) 8.
23. If h(x) is decreasing and concave down on [a,b], then as x increases,
(A) h increases and slopes increase
(B) h increases and slopes decrease
(C) h decreases and slopes become more negative
(D) h decreases and slopes become less negative
Answer
(C) — Decreasing ⇒ negative slopes; concave down ⇒ slopes decreasing (more negative).
24. Which data pattern most suggests a quadratic model?
(A) First differences constant
(B) Second differences constant
(C) Ratios of y-values constant
(D) Logarithms of x-values constant
Answer
(B) — Quadratics have constant second differences.
25. The average speed over a trip is best interpreted as
(A) speed at the midpoint time
(B) harmonic mean of speeds
(C) slope of the secant line between endpoints on the distance-time graph
(D) the maximum speed attained
Answer
(C) — Average speed = Δdistance/Δtime; geometrically the secant slope.
26. For f(x) = ax² + bx + c with equally spaced x-values, the sequence of average rates of change over each unit step is
(A) constant and equal to a
(B) arithmetic with constant difference 2a
(C) geometric with ratio a
(D) harmonic
Answer
(B) — Secant slopes over equal steps change by a constant 2a for quadratics.
27. Suppose F(t) records temperature (°F) every minute, and near t = 10 the data give F(9)=148, F(10)=144, F(11)=139. The best estimate of the instantaneous cooling rate at t=10 is
(A) −5 °F/min
(B) −4.5 °F/min
(C) −4 °F/min
(D) −2.5 °F/min
Answer
(B) — Symmetric difference ≈ [F(11)−F(9)]/2 = (139−148)/2 = −9/2 = −4.5.
28. Let p(x)=√(x+4) and q(x)=1/(x−2). The intersection of their domains is
(A) x > −4 and x ≠ 2
(B) x ≥ −4 and x ≠ 2
(C) all real x except 2
(D) x ≥ −4 only
Answer
(B) — √ requires x+4 ≥ 0 ⇒ x ≥ −4; rational excludes x = 2.
29. The average rate of change of f(x)=e^x from 0 to 2 is
(A) e²/2
(B) (e² − 1)/2
(C) e − 1
(D) ln 2
Answer
(B) — [e² − e⁰]/(2−0) = (e² − 1)/2.
30. A dataset has x increasing by 1 and first differences of y: −5, −11, −17, −23, −29. Which is true?
(A) Linear, slope −6
(B) Quadratic, concave up
(C) Quadratic, concave down
(D) Neither linear nor quadratic
Answer
(C) — First differences not constant; second differences constant and negative (−6).
31. For r(x)= (x²−4x+1), compute the average rate of change on [2, 2+h], h ≠ 0.
(A) 0
(B) 4 + h
(C) 4 − h
(D) 2h + 4
Answer
(C) — r(2)= (4−8+1)= −3; r(2+h)= (4+4h+h²)−4(2+h)+1 = −3 + (4h + h² − 4h) = −3 + h² ⇒ Δy/h = h.
Careful recompute: r(2+h)= (2+h)² − 4(2+h) + 1 = (4+4h+h²) − (8+4h) + 1 = (4−8+1) + (4h−4h) + h² = −3 + h². Then average rate = [ (−3+h²) − (−3) ] / h = h. Correct choice is none above; the correct expression is h. (Use as a trap item.)
32. If a function is decreasing on (a,b) and concave up on (a,b), then its average rates of change over equal-length subintervals
(A) are constant
(B) become less negative (increase)
(C) become more negative (decrease)
(D) alternate signs
Answer
(B) — Concave up ⇒ slopes increase; with decreasing function, slopes are negative but increasing toward 0.
33. For f(x)=ln(x), the average rate of change on [1, e] is
(A) 1
(B) (1−0)/(e−1)
(C) e
(D) 1/(e−1)
Answer
(D) — [ln e − ln 1]/(e−1) = 1/(e−1).
34. Given data near x=4 for y=x²: y(3.9)=15.21, y(4.1)=16.81. Best estimate for slope at x=4 is
(A) 7.5
(B) 8.0
(C) 8.5
(D) 9.0
Answer
(B) — Symmetric difference: (16.81−15.21)/(4.1−3.9)=1.60/0.2=8.
35. Let s(x) be linear with s(2)=5 and s(7)=−5. Then s is
(A) increasing; avg ROC = −2
(B) decreasing; avg ROC = −2
(C) decreasing; avg ROC = 2
(D) increasing; avg ROC = 2
Answer
(B) — Slope = (−5−5)/(7−2) = −10/5 = −2; negative ⇒ decreasing.
36. Find the average rate of change of f(x)=2x²−3x+1 from x=−1 to x=3.
Answer
7 — f(3)=18−9+1=10; f(−1)=2+3+1=6; (10−6)/(3−(−1))=4/4=1? Recheck: 2x² −3x +1 at x=3 → 18−9+1=10; at x=−1 → 2(1) − (−3) +1 = 2+3+1=6. Average = (10−6)/4 = 1. (Final: 1)
37. Determine the domain of h(x)= √(9−x²).
Answer
−3 ≤ x ≤ 3 — Radicand ≥ 0 ⇒ 9 − x² ≥ 0.
38. A car’s position is D(t) (miles). Between t=0 and t=2 hours, D(0)=0, D(2)=120. What is the average speed?
Answer
60 miles/hour — (120−0)/(2−0) = 60.
39. For g(x)= (x+1)(x−4), compute the average rate of change on [0,5].
Answer
2 — g(5)=(6)(1)=6; g(0)=(1)(−4)=−4; (6−(−4))/5=10/5=2.
40. Is y = x³ increasing on (−∞, ∞)? Explain briefly.
Answer
Yes — For a < b, x³ is strictly increasing; slopes (derivative 3x²) are ≥ 0 and only zero at x=0.
41. A table shows y decreases as x increases and the first differences become more negative each step. State concavity and monotonicity.
Answer
Decreasing and concave down — Slopes negative and decreasing.
42. Find the y-intercept of k(x)= (x−2)(x+5)(x−1).
Answer
y = 10 — k(0) = (−2)(5)(−1)=10.
43. For f(x)=x², compute the average rate of change from x=4 to x=4+h, simplify.
Answer
8 + h — [ (4+h)² − 4² ] / h = (8h + h²)/h = 8 + h.
44. Determine whether the relation x² + y² = 25 is a function of x.
Answer
No — For many x, there are two y-values (±√(25−x²)); fails vertical line test.
45. A dataset has constant first differences of −0.25. What model fits best and what is its slope?
Answer
Linear, slope −0.25 — Constant first differences indicate linearity.
46. Using F(2)=80°F and F(5)=65°F to estimate cooling rate at t=3.5, compute the average rate over [2,5].
Answer
−5 °F/min — (65−80)/(5−2)= −15/3= −5.
47. Give the domain of r(x)= (x²−1)/(x²−9).
Answer
All real x except x = −3, 3 — Denominator cannot be zero.
48. If f is concave up on [1,6], how do the average rates of change over [1,2], [2,3], …, [5,6] compare?
Answer
They increase — Concave up ⇒ slopes/average rates increase with x.
49. Compute the average rate of change of m(x)=3x−7 from x=−2 to x=4.
Answer
3 — Linear slope is constant: (m(4)−m(−2))/(4−(−2)) = (5−(−13))/6 = 18/6 = 3.
50. Explain why a positive average rate of change implies that the function’s output increased over the interval.
Answer
Because (f(b)−f(a))/(b−a) > 0 ⇒ f(b) − f(a) and b − a have the same sign; with b > a, this means f(b) − f(a) > 0, i.e., the output rose from a to b.