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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 2: Differentiation Definition & Fundamental PropertiesDefinition of Differentiation

Definition of Differentiation (College Board AP® Calculus AB)

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Definition of Differentiation

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Definition of Differentiation

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1
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Let f be the function given by f open parentheses x close parentheses equals 1 over x and let g be the function given by g open parentheses x close parentheses equals negative 11 x squared.

At what value of x do the graphs of f and g have parallel tangent lines?

  • 0.213

  • -0.357

  • 0.357

  • -0.450

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2
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f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator x squared minus 9 over denominator x minus 3 end fraction comma space space space x not equal to 3 end cell row cell 5 comma space space space x equals 3 end cell end table close

Let f be the function defined above. Which of the following statements about f are true?

I. f has a limit at x equals 3.

II. f is continuous at x equals 3.

III. f is differentiable at x equals 3.

  • I only

  • II only

  • III only

  • I, II, and III

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3
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At x equals 2, the function given by f open parentheses x close parentheses equals open curly brackets table row cell 10 x minus 25 comma space space space x less or equal than 5 end cell row cell x squared comma space space space x greater than 5 end cell end table close is

  • Continuous but not differentiable

  • Differentiable but not continuous

  • Neither continuous nor differentiable

  • Both continuous and differentiable

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4
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Graph of function f with points labelled along the x-axis: a, b, c and d.

The graph of a function f is shown above. At which value of x is f continuous but not differentiable?

  • a

  • b

  • c

  • d

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5
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If the line tangent to the graph of the function f at the point open parentheses 2 comma space 9 close parentheses passes through the point open parentheses negative 1 comma space 3 close parentheses, then f apostrophe open parentheses 2 close parentheses is

  • negative 2

  • 2

  • 3

  • 12

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1
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Let f be the function defined by f open parentheses x close parentheses equals square root of open vertical bar x minus 5 close vertical bar end root for all x. Which of the following statements is true?

  • f is continuous but not differentiable at x equals 5.

  • f is differentiable at x equals 5.

  • f is not continuous at x equals 5.

  • limit as x rightwards arrow 5 of f open parentheses x close parentheses not equal to 0

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2
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limit as h rightwards arrow 0 of fraction numerator sin open parentheses pi plus h close parentheses minus sin open parentheses pi close parentheses over denominator h end fraction is

  • negative 1

  • 0

  • 1 half

  • 1

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3
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negative quadratic from -1,2 to 1, 2 with open circle. Closed circle at 1,0. Semi circular section between 1,0 and 3,0, and a larger semicircle between 3,0 and 7,0

The graph of the function f shown in the figure above has a vertical tangent at the point (3, 0) and horizontal tangents at the points (2, 0) and (5, -2).

For what values of x, negative 1 less than x less than 7 is f not differentiable?

  • 1 only

  • 1 and 3 only

  • 2 and 5 only

  • 1, 2, and 5 only

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4
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Let f be the function defined by f left parenthesis x right parenthesis equals 2 x cubed plus 4 x minus 1. Which of the following is an equation of the line tangent to the graph of f at the point where x equals 1?

  • y equals 10 x minus 5

  • y equals 10 x plus 15

  • y equals 4 x plus 1

  • y equals 4 x plus 9

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5
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Let f be a differentiable function such that f open parentheses 10 close parentheses equals 8 and f apostrophe open parentheses x close parentheses less or equal than 2 for all x. Of the following, which is not a possible value for f open parentheses 6 close parentheses?

  • negative 3

  • 0

  • 6

  • 9

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1
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f open parentheses x close parentheses equals open curly brackets table row cell 5 minus x squared comma space space space x less than negative 3 end cell row cell 6 over x minus 1 comma space space space minus 3 less or equal than x less than negative 1 end cell row cell 2 x minus 5 comma space space space x greater or equal than negative 1 end cell end table close

Let f be the function defined above. At what values of x, if any, is f not differentiable?

  • x equals negative 3 only

  • x equals negative 1 only

  • x equals negative 3 and x equals negative 1

  • f is differentiable for all values of x

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2
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Which of the following is an equation of the tangent line to the graph of f open parentheses x close parentheses equals 2 x to the power of 4 plus 3 x squared at the point where f to the power of apostrophe open parentheses x close parentheses equals negative 1?

  • y equals x plus 0.082

  • y equals negative x minus 0.161

  • y equals negative x plus 0.079

  • y equals negative x minus 0.082

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3
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f open parentheses x close parentheses equals open curly brackets table row cell a x minus b comma space space space x less or equal than 2 end cell row cell x squared plus open parentheses 10 minus a close parentheses x comma space space space x greater than 2 end cell end table close

Let f be the function defined above, where a and b are constants. If f is differentiable at x equals 2, what is the value of a minus b?

  • 2

  • 3

  • 4

  • 7

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4
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In the x y-plane, the line 8 x plus y equals k, where k is a constant, is tangent to the graph with derivative fraction numerator d y over denominator d x end fraction equals 6 x minus 2 at open parentheses x comma space 10 close parentheses. What is the value of k?

  • 0

  • 1

  • 2

  • 3

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5
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A differentiable function f has the property that f apostrophe open parentheses x close parentheses less than negative 1 for negative 2 less or equal than x less or equal than 7 and f open parentheses 6 close parentheses equals negative 4. Which of the following could be true?

I. f open parentheses 3 close parentheses equals negative 5

II. f open parentheses negative 1 close parentheses equals 3

III. f open parentheses 2 close parentheses equals 4

  • I only

  • II only

  • III only

  • I, II and III

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