Calculus Practice Problems
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Questions and Answers

What is the nature of the function f5 on the interval [−5, −3]?

  • Constant
  • Strictly decreasing
  • Neither strictly increasing nor decreasing
  • Strictly increasing (correct)

    • What is the x-value of the strict local minimum of the function f6?

  • 6
  • 3
  • −3
  • 2 (correct)

    • What is the nature of the function f7 on the interval (−∞, 8)?

  • Constant
  • Strictly decreasing (correct)
  • Neither strictly increasing nor decreasing
  • Strictly increasing

    • What is the nature of the function f8 on the interval (−∞, 1)?

    <p>Strictly increasing</p> Signup and view all the answers

    What is the nature of the function f1?

    <p>Weakly concave up and weakly concave down</p> Signup and view all the answers

    What is the nature of the function f2?

    <p>Strictly concave up</p> Signup and view all the answers

    What is the nature of the function f3 on the interval (−∞, 7)?

    <p>Strictly concave down</p> Signup and view all the answers

    What is the x-value of the inflection point of the function f3?

    <p>7</p> Signup and view all the answers

    What is the nature of the function f4?

    <p>The second derivative is a linear function</p> Signup and view all the answers

    What is the nature of the function f8 on the interval (1, +∞)?

    <p>Strictly decreasing</p> Signup and view all the answers

    Study Notes

    Derivatives and Tangent Lines

    • f1(x) = 35x - 6, f1'(x) = 35, f1'(1) = 23, L1(x) = 23(x - 1) - 7
    • f2(x) = -6x^4, f2'(-2) = -8, L2(x) = 8(x + 2) - 4
    • f3(x) = √x, f3(1/4) = 2, L3(x) = 2x - 4 + 1
    • f4(x) = -12x + 34, f4'(2) = 10, L4(x) = 10(x - 2) + 4
    • f5(x) = 2x^2 + 16x + 32, f5'(-3) = -13, L5(x) = -13(x + 3) + 4
    • f6(x) = -6 ln(7) * 7^x, f6(0) = -6 ln(7), L6(x) = -6 ln(7) x + 41
    • f7(x) = -12(2 - 3x)^3, f7'(2) = 768, L7(x) = 768(x - 2) + 256
    • f8(x) = (2x - 2)e^(x - 2)^(-15), f8'(-3) = -8, L8(x) = -8(x + 3) + 1
    • f9(x) = 2e^(2x) ln(3x - 5) + e^(2x) / (3x - 5), f9'(2) ≈ 163.79, L9(x) = 163.79(x - 2)
    • f10(x) = (10x + 6x)(8x^3 - 9x^2 + 100) + (5x^2 + 6x - 7)(24x^2 - 18x), f10'(-2) = 132, L10(x) = 132(x + 2)

    Monotonicity and Extreme Values

    • f1(x) = -8x, f1'(x) = -8 < 0 everywhere, f1 strictly decreases everywhere
    • f2(x) = x^2 - 6x + 42, f2'(x) = 2x - 6, f2 strictly decreases on (-∞, 3], strictly increases on [3, +∞)
    • f3(x) = x^3 - 243x, f3'(x) = 3x^2 - 243, strictly increases on (-∞, -9], strictly decreases on [-9, 9], strictly increases on [9, +∞)
    • f4(x) = x^3 + 3x^2 + 3x + 100, f4'(x) = 3x^2 + 6x + 3 ≥ 0, f4 strictly increases on ℝ
    • f5(x) = -3x^4 - 32x^3 - 90x^2, f5'(x) = -12x^3 - 96x^2 - 180x, f5 strictly decreases on ℝ

    Convexity and Inflection Points

    • f1(x) = 21/x + 7, f1''(x) = 0, f1 is everywhere concave up and concave down
    • f2(x) = x^2 - 6x + 42, f2''(x) = 2 > 0, f2 is strictly concave up everywhere
    • f3(x) = x^3 - 21x^2, f3''(x) = 6x - 42, f3 is strictly concave down on (-∞, 7], strictly concave up on [7, +∞), x = 7 is an inflection point
    • f4(x) = x^3 + 3x^2 + 3x + 100, f4''(x) = 6x + 6

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