Calculus Quiz - Non Calculator and Calculator Sections
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Questions and Answers

What is the domain of the function f(x) if it is defined as f(x) = undefined?

  • f(x) exists for all real numbers
  • x < -2 or x > 2 (correct)
  • x = -2 or x = 2
  • -2 < x < 2

    • If g(x) is such that g'(4) = -24, what can you infer about g(4)?

  • g(4) is constant.
  • g(4) is increasing.
  • g(4) is undefined.
  • g(4) is decreasing. (correct)

    • What value of k will make the function f(x) continuous for all x?

  • There is no real value of k that makes f(x) continuous for all x. (correct)
  • 16
  • 0
  • Any real number

    • For the function f(x) = 2x² - 7x - 10, what is the absolute maximum value of f(x) on the interval [-1, 3]?

    <p>7/4</p> Signup and view all the answers

    What is the acceleration of a particle described by the function s = t³ - 6t² + 9t at time t = 4?

    <p>0</p> Signup and view all the answers

    What is the equation of the tangent line to the curve 9x² + 16y² = 52 at the point (2, -1)?

    <p>9x - 8y - 26 = 0</p> Signup and view all the answers

    If f(x) = some expression, then f'(x) is most likely related to the:

    <p>Original function f(x)</p> Signup and view all the answers

    According to the Mean Value Theorem, what is the value c that satisfies the conclusion for f(x) on the interval [2, 5]?

    <p>A value not within the interval</p> Signup and view all the answers

    What is the correct value of f'(x) if f(x) = 3?

    <p>0</p> Signup and view all the answers

    What is the third derivative f'''(x) of the function f(x) = sin(2x)?

    <p>-4 sin(2x)</p> Signup and view all the answers

    What is the slope of the normal line to y = x + cos(xy) at the point (0, 1)?

    <p>-1</p> Signup and view all the answers

    If a 20-foot ladder is sliding down a wall at 5 ft/sec and the top is 10 feet from the floor, how fast is the bottom sliding out?

    <p>0.346</p> Signup and view all the answers

    How fast is the distance between Boat A and Boat B increasing after 2.5 hours if Boat A heads North at 12 km/hr and Boat B heads East at 18 km/hr?

    <p>31.20</p> Signup and view all the answers

    What is the approximate change in the volume of a sphere when the radius increases from 10 to 10.02 cm?

    <p>1256.637</p> Signup and view all the answers

    What is the value of f’(8) if f(x) is defined but not given?

    <p>40</p> Signup and view all the answers

    If f(x) is continuous for all real numbers and f(4) = 0, what can be inferred about its continuity at 4?

    <p>f(4) is continuous and equals 0</p> Signup and view all the answers

    What is the condition for continuity at the point where f(x) is differentiable if the value of b is what?

    <p>0</p> Signup and view all the answers

    What is the local minimum point of the function given in the graph?

    <p>(1.66, -0.59)</p> Signup and view all the answers

    What can be concluded if f(x) is differentiable everywhere?

    <p>f(x) is continuous everywhere</p> Signup and view all the answers

    If f(x) = sec x + csc x, what is the correct expression for f’(x)?

    <p>sec x tan x - csc x cot x</p> Signup and view all the answers

    An equation of the line normal to the graph of y = 3x at the point (2, 4) can be categorized as:

    <p>y - 4 = -3(x - 2)</p> Signup and view all the answers

    For which value of x does f(x) = x^2 - 4x + 4 have a local minimum?

    <p>2</p> Signup and view all the answers

    If f(x) = x^3 - 6x^2 + 9x, which of the following statements is true?

    <p>f(x) is continuous everywhere</p> Signup and view all the answers

    If f(x) represents a continuous function without breaks, which conclusions are definite?

    <p>f(x) is defined everywhere</p> Signup and view all the answers

    Study Notes

    Multiple Choice Questions - Non Calculator Section

    • Question 1: If f(x) = 5x³, then f'(8) = 40
    • Question 2: lim (5x² - 3x + 1) / (4x² + 2x + 5) as x approaches infinity is 5/4
    • Question 3: If f(x) = (3x² + x) / (3x² - x), then f'(x) = (6x² + 1) / (3x² - x)²
    • Question 4: If the function f is continuous for all real numbers and f(x) = (x² - 7x + 12) / (x - 4) when x ≠ 4, then f(4) = 5
    • Question 5: If x² - 2xy + 3y² = 8, then dy/dx = (2x - 2y) / (6y - 2x).
    • Question 6: If f(x) = sec x + csc x, then f'(x) = sec x tan x + csc x cot x
    • Question 7: An equation of the line normal to the graph of y = √(3x² + 2x) at (2, 4) is 4x+7y=36
    • Question 8: If f(x) = cos²x, then f" (π) = -2.
    • Question 9: If f(x) = x² + 1 and g(x) = 3x, then g(f(2)) = 37
    • Question 10: The slope of the line tangent to the graph of 3x² + 5 ln y = 12 at (2, 1) is 12/5

    Multiple Choice Questions - Calculator Section

    • Question 11: If f(x) is continuous everywhere for all real numbers x, which of the following must be true?

    • I. f(x) is continuous everywhere

    • II. f(x) is differentiable everywhere

    • III. f(x) has a local minimum at x = 2

    • Answer: I only

    • Question 12: For what value of x does the function f(x) = x³ - 9x² - 120x + 6 have a local minimum?

    • Answer: 10

    • Question 13: lim (sin x cos x - sin x) / x as x approaches 0 is 1

    • Question 14: If f(x) = cos(x + 1), then f' (π) = -3cos²(π +1)sin(n + 1)

    • Question 15: lim (tan(π/6 + h) - tan(π/6)) / h as h approaches 0 is √3

    • Question 16: If g(x) = 3x⁴ - 5x², find g'(4) is -72

    • Question 17: The domain of the function f(x) = √(4 - x²) is -2 ≤ x ≤ 2

    • Question 18: lim (x² - 25) / (x - 5) as x approaches 5 is 10

    • Question 19: Evaluate lim h→0 (5/(5+h)^2 - 5/25) / h is 2

    • Question 20: Find k so that f(x) = (x² - 16) / (x - 4); x ≠ 4, k ; x = 4 is continuous for all x.

    • Answer: k=8

    • Question 21: If f(x) = x² cos 2x, find f '(x).

    • Answer:- 2x cos 2x + 2x²sin 2x

    • Question 22: An equation of the line tangent to y= 4x³ - 7x² at x = 3 is y - 45 = 66(x- 3)

    • Question 23: Find a positive value c for x that satisfies the conclusion of the Mean Value Theorem for Derivatives for f(x) = 3x² − 5x + 1 on the interval [2, 5].

    • Answer: 11/6

    • Question 24: Given f(x) = 2x² - 7x - 10, find the absolute maximum of f(x) on [-1, 3].

    • Answer: -8

    • Question 25: Find dy/dx if x²y + xy² = -10.

    • Answer: (3x²y+y³) / (3x² + 3xy²)

    • Question 26: Find the equation of the tangent line to 9x² + 16y² = 52 through (2, -1).

    • Answer: 9x-8y-26=0

    • Question 27: A particle's position is given by s= t³ - 6t² + 9t. What is its acceleration at time t = 4?

    • Answer: 12

    • Question 28: If f(x) = 3ˣ, then f'(x) = 3ˣ ln 3

    • Question 29: If f(x) = sin²x, find f''(x).

    • Answer: - 4 sin x cos x

    • Question 30: Find the slope of the normal line to y = x + cos xy at (0, 1).

    • Answer: -1

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    Description

    Test your understanding of calculus concepts with this quiz, which includes both non-calculator and calculator sections. Questions cover derivatives, limits, continuity, and tangent lines. Perfect for students looking to reinforce their skills in calculus fundamentals.

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