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AP Precalculus Flashcards: Linear And Quadratic Rates Of Change

Study Linear And Quadratic Rates Of Change in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Linear And Quadratic Rates Of Change

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QUESTION
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Find the average rate of change of f(x)=x2+4xf(x) = x^2 + 4x over [3,6][3, 6].

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ANSWER
  1. Calculate f(6)−f(3)6−3=60−213=13\frac{f(6) - f(3)}{6 - 3} = \frac{60 - 21}{3} = 13.

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What this deck covers

This deck focuses on Linear And Quadratic Rates Of Change, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

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Flashcard 1: Find the average rate of change of f(x)=x2+4xf(x) = x^2 + 4x over [3,6][3, 6].

Answer:

  1. Calculate f(6)−f(3)6−3=60−213=13\frac{f(6) - f(3)}{6 - 3} = \frac{60 - 21}{3} = 13.

Flashcard 2: What is the derivative of f(x)=x2−4x+2f(x) = x^2 - 4x + 2?

Answer: 2x−42x - 4. Apply power rule and linear term differentiation.

Flashcard 3: What is the instantaneous rate of change of f(x)=2x2f(x) = 2x^2 at x=−1x = -1?

Answer: -4. Find f′(x)=4xf'(x) = 4x, then evaluate at x=−1x = -1.

Flashcard 4: State the formula for finding the slope of a secant line over [a,b][a, b].

Answer: f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}. Same formula as average rate of change between two points.

Flashcard 5: Find the average rate of change of f(x)=x2+2x−1f(x) = x^2 + 2x - 1 over [2,4][2, 4].

Answer:

  1. Calculate f(4)−f(2)4−2=23−72=8\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8.

Flashcard 6: State the formula for the derivative of f(x)=ax2+bx+cf(x) = ax^2 + bx + c.

Answer: 2ax+b2ax + b. General form of derivative for quadratic functions.

Flashcard 7: What is the instantaneous rate of change of f(x)=x3f(x) = x^3 at x=1x = 1?

Answer:

  1. Find f′(x)=3x2f'(x) = 3x^2, then evaluate at x=1x = 1.

Flashcard 8: Find the average rate of change of f(x)=x2+2x−1f(x) = x^2 + 2x - 1 over [2,4][2, 4].

Answer:

  1. Calculate f(4)−f(2)4−2=23−72=8\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8.

Flashcard 9: Identify the instantaneous rate of change of f(x)=x2f(x) = x^2 at x=3x = 3.

Answer:

  1. Find f′(x)=2xf'(x) = 2x, then evaluate at x=3x = 3.

Flashcard 10: What is the formula for the average rate of change of a function f(x)f(x) over [a,b][a, b]?

Answer: f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}. This is the slope formula for secant lines connecting two points.

Flashcard 11: What is the average rate of change of f(x)=−x2+2xf(x) = -x^2 + 2x over [1,5][1, 5]?

Answer: -4. Calculate f(5)−f(1)5−1=−15−14=−4\frac{f(5) - f(1)}{5 - 1} = \frac{-15 - 1}{4} = -4.

Flashcard 12: Identify the average rate of change of f(x)=6x−x2f(x) = 6x - x^2 over [0,3][0, 3].

Answer:

  1. Calculate f(3)−f(0)3−0=9−03=3\frac{f(3) - f(0)}{3 - 0} = \frac{9 - 0}{3} = 3.

Flashcard 13: State the formula for finding the slope of a secant line over [a,b][a, b].

Answer: f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}. Same formula as average rate of change between two points.

Flashcard 14: State the derivative of f(x)=x2+5x−3f(x) = x^2 + 5x - 3.

Answer: 2x+52x + 5. Differentiate each term using power rule and constant rule.

Flashcard 15: Identify the instantaneous rate of change of f(x)=x3−3x2f(x) = x^3 - 3x^2 at x=2x = 2.

Answer:

  1. Find f′(x)=3x2−6xf'(x) = 3x^2 - 6x, then substitute x=2x = 2.

Flashcard 16: What is the average rate of change of f(x)=x2f(x) = x^2 over [1,3][1, 3]?

Answer:

  1. Calculate f(3)−f(1)3−1=9−12=4\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4.

Flashcard 17: Find the slope of the tangent line to f(x)=x2−5x+6f(x) = x^2 - 5x + 6 at x=3x = 3.

Answer:

  1. Find f′(x)=2x−5f'(x) = 2x - 5, then substitute x=3x = 3.

Flashcard 18: Find the slope of the tangent line to f(x)=4x2−xf(x) = 4x^2 - x at x=1x = 1.

Answer:

  1. Find f′(x)=8x−1f'(x) = 8x - 1, then evaluate at x=1x = 1.

Flashcard 19: Identify the expression for the derivative of f(x)=x3−2xf(x) = x^3 - 2x.

Answer: 3x2−23x^2 - 2. Apply power rule to each term: (x3)′=3x2(x^3)' = 3x^2 and (2x)′=2(2x)' = 2.

Flashcard 20: Find the average rate of change of f(x)=5x2+2xf(x) = 5x^2 + 2x over [1,3][1, 3].

Answer:

  1. Calculate f(3)−f(1)3−1=51−72=22\frac{f(3) - f(1)}{3 - 1} = \frac{51 - 7}{2} = 22.

Flashcard 21: State the formula for the derivative of f(x)=3x2+4x+5f(x) = 3x^2 + 4x + 5.

Answer: 6x+46x + 4. Differentiate each term using power rule and sum rule.

Flashcard 22: What is the derivative of f(x)=5x2−3x+1f(x) = 5x^2 - 3x + 1?

Answer: 10x−310x - 3. Differentiate using power rule on each term.

Flashcard 23: Find the average rate of change of f(x)=x2+4xf(x) = x^2 + 4x over [3,6][3, 6].

Answer:

  1. Calculate f(6)−f(3)6−3=60−213=13\frac{f(6) - f(3)}{6 - 3} = \frac{60 - 21}{3} = 13.

Flashcard 24: Identify the derivative of f(x)=2x3−x2+4f(x) = 2x^3 - x^2 + 4.

Answer: 6x2−2x6x^2 - 2x. Apply power rule: (2x3)′=6x2(2x^3)' = 6x^2 and (−x2)′=−2x(-x^2)' = -2x.

Flashcard 25: Identify the instantaneous rate of change of f(x)=x3−3x2f(x) = x^3 - 3x^2 at x=2x = 2.

Answer:

  1. Find f′(x)=3x2−6xf'(x) = 3x^2 - 6x, then substitute x=2x = 2.

Flashcard 26: State the derivative of f(x)=x2+5x−3f(x) = x^2 + 5x - 3.

Answer: 2x+52x + 5. Differentiate each term using power rule and constant rule.

Flashcard 27: Find the slope of the tangent line to f(x)=x2+3xf(x) = x^2 + 3x at x=2x = 2.

Answer:

  1. Find f′(x)=2x+3f'(x) = 2x + 3, then substitute x=2x = 2.

Flashcard 28: State the formula for the derivative of f(x)=ax2+bx+cf(x) = ax^2 + bx + c.

Answer: 2ax+b2ax + b. General form of derivative for quadratic functions.

Flashcard 29: What is the derivative of f(x)=3x2+2x+1f(x) = 3x^2 + 2x + 1?

Answer: 6x+26x + 2. Apply power rule: (xn)′=nxn−1(x^n)' = nx^{n-1} and sum rule.

Flashcard 30: State the relationship between secant and tangent lines in context of rates of change.

Answer: Secant: average rate, Tangent: instantaneous rate. Secant connects two points; tangent touches at one point.

Flashcard 31: What is the derivative of f(x)=−x2+3xf(x) = -x^2 + 3x?

Answer: −2x+3-2x + 3. Apply power rule and sum rule to find the derivative.

Flashcard 32: What is the derivative of f(x)=3x2+2x+1f(x) = 3x^2 + 2x + 1?

Answer: 6x+26x + 2. Apply power rule: (xn)′=nxn−1(x^n)' = nx^{n-1} and sum rule.

Flashcard 33: Find the slope of the tangent line to f(x)=x2−5x+6f(x) = x^2 - 5x + 6 at x=3x = 3.

Answer:

  1. Find f′(x)=2x−5f'(x) = 2x - 5, then substitute x=3x = 3.

Flashcard 34: Identify the average rate of change of f(x)=3x−4f(x) = 3x - 4 over [2,5][2, 5].

Answer:

  1. Linear functions have constant rate equal to the coefficient of xx.

Flashcard 35: What is the instantaneous rate of change of f(x)=4x2f(x) = 4x^2 at x=2x = 2?

Answer:

  1. Find f′(x)=8xf'(x) = 8x, then evaluate at x=2x = 2.

Flashcard 36: Identify the average rate of change of f(x)=2x+3f(x) = 2x + 3 over [1,4][1, 4].

Answer:

  1. Linear functions have constant slope, which equals the coefficient of xx.

Flashcard 37: Identify the average rate of change of f(x)=−x2+4f(x) = -x^2 + 4 over [0,2][0, 2].

Answer: -2. Calculate f(2)−f(0)2−0=0−42=−2\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 4}{2} = -2.

Flashcard 38: State the relationship between secant and tangent lines in context of rates of change.

Answer: Secant: average rate, Tangent: instantaneous rate. Secant connects two points; tangent touches at one point.

Flashcard 39: Identify the instantaneous rate of change of f(x)=x2f(x) = x^2 at x=3x = 3.

Answer:

  1. Find f′(x)=2xf'(x) = 2x, then evaluate at x=3x = 3.

Flashcard 40: Identify the slope of the tangent line to f(x)=x3−3xf(x) = x^3 - 3x at x=1x = 1.

Answer:

  1. Find f′(x)=3x2−3f'(x) = 3x^2 - 3, then evaluate at x=1x = 1.

Flashcard 41: State the formula for the derivative of f(x)=3x2+4x+5f(x) = 3x^2 + 4x + 5.

Answer: 6x+46x + 4. Differentiate each term using power rule and sum rule.

Flashcard 42: What is the average rate of change of f(x)=4x−7f(x) = 4x - 7 over [0,2][0, 2]?

Answer: 44. Linear functions have constant rate of change equal to the slope.

Flashcard 43: What is the instantaneous rate of change of f(x)=x2+2xf(x) = x^2 + 2x at x=0x = 0?

Answer:

  1. Find f′(x)=2x+2f'(x) = 2x + 2, then substitute x=0x = 0.

Flashcard 44: What is the instantaneous rate of change of f(x)=x2+2xf(x) = x^2 + 2x at x=0x = 0?

Answer:

  1. Find f′(x)=2x+2f'(x) = 2x + 2, then substitute x=0x = 0.

Flashcard 45: What is the derivative of f(x)=x2−4x+2f(x) = x^2 - 4x + 2?

Answer: 2x−42x - 4. Apply power rule and linear term differentiation.

Flashcard 46: What is the instantaneous rate of change of f(x)=3x2−5xf(x) = 3x^2 - 5x at x=4x = 4?

Answer:

  1. Find f′(x)=6x−5f'(x) = 6x - 5, then evaluate at x=4x = 4.

Flashcard 47: Identify the average rate of change of f(x)=−x2+4f(x) = -x^2 + 4 over [0,2][0, 2].

Answer: -2. Calculate f(2)−f(0)2−0=0−42=−2\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 4}{2} = -2.

Flashcard 48: Find the average rate of change of f(x)=5x2+2xf(x) = 5x^2 + 2x over [1,3][1, 3].

Answer:

  1. Calculate f(3)−f(1)3−1=51−72=22\frac{f(3) - f(1)}{3 - 1} = \frac{51 - 7}{2} = 22.

Flashcard 49: What is the instantaneous rate of change of f(x)=2x2f(x) = 2x^2 at x=−1x = -1?

Answer: −4-4. Find f′(x)=4xf'(x) = 4x, then evaluate at x=−1x = -1.

Flashcard 50: What is the derivative of f(x)=5x2−3x+1f(x) = 5x^2 - 3x + 1?

Answer: 10x−310x - 3. Differentiate using power rule on each term.

Flashcard 51: What is the derivative of f(x)=−x2+3xf(x) = -x^2 + 3x?

Answer: −2x+3-2x + 3. Apply power rule and sum rule to find the derivative.

Flashcard 52: Identify the average rate of change of f(x)=2x+3f(x) = 2x + 3 over [1,4][1, 4].

Answer:

  1. Linear functions have constant slope, which equals the coefficient of xx.

Flashcard 53: What is the derivative of f(x)=7x2f(x) = 7x^2?

Answer: 14x14x. Use power rule on 7x27x^2 to get 7⋅2x=14x7 \cdot 2x = 14x.

Flashcard 54: What is the instantaneous rate of change of f(x)=x2−xf(x) = x^2 - x at x=0x = 0?

Answer: -1. Find f′(x)=2x−1f'(x) = 2x - 1, then substitute x=0x = 0.

Flashcard 55: Identify the average rate of change of f(x)=6x−x2f(x) = 6x - x^2 over [0,3][0, 3].

Answer:

  1. Calculate f(3)−f(0)3−0=9−03=3\frac{f(3) - f(0)}{3 - 0} = \frac{9 - 0}{3} = 3.

Flashcard 56: What is the instantaneous rate of change of f(x)=x2−xf(x) = x^2 - x at x=0x = 0?

Answer: -1. Find f′(x)=2x−1f'(x) = 2x - 1, then substitute x=0x = 0.

Flashcard 57: What is the instantaneous rate of change of f(x)=3x2−5xf(x) = 3x^2 - 5x at x=4x = 4?

Answer:

  1. Find f′(x)=6x−5f'(x) = 6x - 5, then evaluate at x=4x = 4.

Flashcard 58: What is the average rate of change of f(x)=−x2+2xf(x) = -x^2 + 2x over [1,5][1, 5]?

Answer: -4. Calculate f(5)−f(1)5−1=−15−14=−4\frac{f(5) - f(1)}{5 - 1} = \frac{-15 - 1}{4} = -4.

Flashcard 59: What is the formula for the average rate of change of a function f(x)f(x) over [a,b][a, b]?

Answer: f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}. This is the slope formula for secant lines connecting two points.

Flashcard 60: Identify the expression for the derivative of f(x)=x3−2xf(x) = x^3 - 2x.

Answer: 3x2−23x^2 - 2. Apply power rule to each term: (x3)′=3x2(x^3)' = 3x^2 and (2x)′=2(2x)' = 2.

Flashcard 61: State the formula for the instantaneous rate of change of f(x)f(x) at x=ax = a.

Answer: f′(a)f'(a). The derivative gives the instantaneous rate of change at any point.

Flashcard 62: What is the instantaneous rate of change of f(x)=4x2f(x) = 4x^2 at x=2x = 2?

Answer:

  1. Find f′(x)=8xf'(x) = 8x, then evaluate at x=2x = 2.

Flashcard 63: Identify the average rate of change of f(x)=3x−4f(x) = 3x - 4 over [2,5][2, 5].

Answer:

  1. Linear functions have constant rate equal to the coefficient of xx.

Flashcard 64: What is the derivative of f(x)=7x2f(x) = 7x^2?

Answer: 14x14x. Use power rule on 7x27x^2 to get 7⋅2x=14x7 \cdot 2x = 14x.

Flashcard 65: State the formula for the instantaneous rate of change of f(x)f(x) at x=ax = a.

Answer: f′(a)f'(a). The derivative gives the instantaneous rate of change at any point.

Flashcard 66: Identify the slope of the tangent line to f(x)=x3−3xf(x) = x^3 - 3x at x=1x = 1.

Answer:

  1. Find f′(x)=3x2−3f'(x) = 3x^2 - 3, then evaluate at x=1x = 1.

Flashcard 67: What is the instantaneous rate of change of f(x)=x3f(x) = x^3 at x=1x = 1?

Answer:

  1. Find f′(x)=3x2f'(x) = 3x^2, then evaluate at x=1x = 1.

Flashcard 68: What is the average rate of change of f(x)=x2f(x) = x^2 over [1,3][1, 3]?

Answer:

  1. Calculate f(3)−f(1)3−1=9−12=4\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4.

Flashcard 69: Identify the derivative of f(x)=2x3−x2+4f(x) = 2x^3 - x^2 + 4.

Answer: 6x2−2x6x^2 - 2x. Apply power rule: (2x3)′=6x2(2x^3)' = 6x^2 and (−x2)′=−2x(-x^2)' = -2x.

Flashcard 70: Find the slope of the tangent line to f(x)=x2+3xf(x) = x^2 + 3x at x=2x = 2.

Answer:

  1. Find f′(x)=2x+3f'(x) = 2x + 3, then substitute x=2x = 2.