Differentiability and the Theorems Behind the Rules · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Where the tangent breaks down

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. For x not equal to a, which expression equals f(x) minus f(a)?

    • The quotient plus x minus a
    • The difference quotient times x minus a
    • The difference quotient divided by x minus a
    • The output change times x minus a

  2. If f is differentiable at the point a, which statement must be true?

    • f is continuous at a
    • f has a corner at a
    • f is constant near a

  3. Which function is continuous at 0 but not differentiable at 0?

    • f(x) = x squared
    • f(x) = |x|
    • f(x) = 2x plus 1

  4. Priya claims: if f is differentiable at a, then the limit of f(x) as x tends to a equals f(a). Is Priya right?

    Circle one:   True   False

  5. The mean value theorem requires differentiability at both endpoints. Is this true?

    Circle one:   True   False

  6. Let f(x) equal x squared on [0, 4]. The mean value theorem gives c with f prime of c equal to the secant slope. What is c? Give a decimal.

    Answer: ______________

  7. Sam says every continuous function is differentiable at every point. Is Sam right?

    Circle one:   True   False

  8. Differentiability at a requires f to be differentiable at every nearby point. Is this true?

    Circle one:   True   False

  9. A student proves differentiability implies continuity by writing the output change as quotient times input change. A friend objects that the quotient might not exist. What settles it?

    • Continuity at a must be assumed before the proof begins
    • Differentiability means the quotient limit exists
    • The input change x minus a is always exactly zero

  10. Let f(x) be the square root of x on [1, 9]. The mean value theorem gives c with f'(c) equal to the secant slope. What is c? Give a decimal.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Where the tangent breaks down W1-mt_1xLEeoA69z-s1

  1. The difference quotient times x minus a · For x not equal to a, the output change is the difference quotient multiplied by x minus a.
  2. f is continuous at a · Differentiability at a makes the difference quotient approach a finite number. Multiplying it by x minus a, which tends to zero, makes f(x) approach f(a).
  3. f(x) = |x| · The absolute value has a corner at zero where secant limits disagree.
  4. True · That limit equation is exactly continuity, which differentiability supplies.
  5. False · The mean value theorem requires differentiability on the open interval, not at its endpoints.
  6. 2 · The secant slope is 4 and f prime is 2x, so c is 2.
  7. False · The claim is false. The absolute value function is continuous at 0, but its left and right secant slopes have different limits there.
  8. False · Differentiability at one point only gives the derivative limit at that point. It does not require differentiability at every nearby point.
  9. Differentiability means the quotient limit exists · Differentiability is defined by that limit existing, so the move is legal.
  10. 4 · The secant slope is (3 minus 1) divided by (9 minus 1), which is 0.25. Since f'(x) = 1 divided by (2 times the square root of x), setting this equal to 0.25 at x = c makes the square root of c equal 2, so c = 4.
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