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AP Calculus AB: 8.4.1 Graphs of Polynomial Functions

Mathematics9 CardsCreated 3 months ago

This flashcard set teaches how to graph polynomial functions using derivatives. It walks through identifying critical points, relative extrema, concavity, and inflection points using the first and second derivatives. These techniques allow for accurate graphing without relying on calculators, based purely on calculus-based analysis.

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Graphs of Polynomial Functions

• To graph a function:
1. Find critical points using the first derivative.
2. Determine where the function is increasing or decreasing.
3. Find inflection points using the second derivative.
4. Determine where the function is concave up or concave down.
• On an interval, the sign of the first derivative indicates whether the function is incr

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Key Terms

Term
Definition

Graphs of Polynomial Functions

• To graph a function:
1. Find critical points using the first derivative.
2. Determine where the function is increasing or decreasing.
3....

note

  • Sketching an accurate drawing of a function requires several steps, even for a polynomial function.

  • First, you need to take ...

Use calculus to determine which of the following is the graph of f (x) = x^ 4 − 8x ^3 + 18x ^2 − 16x + 5.

Use your knowledge of relative extrema, critical points, concavity, points of inflection, and anything else (but a graphing calculator) to help you...

Use calculus to determine which of the following is the graph of
f(x)=2/3x^3+2x^2−30x.

Use your knowledge of relative extrema, critical points, concavity, points of inflection, and anything else (but a graphing calculator) to help you...

Which of the following graphs describes a function with these characteristics?
f ′(−2) = 0
f ′(2) = 0
f ″(−2) = 0
f ″(0) = 0
f ″(2) = 0
f ″(x) > 0 when x < −2
f ″(x) < 0 when −2 < x < 0
f ″(x) > 0 when 0 < x < 2
f ″(x) < 0 when x > 2

The graph of f (x) has a horizontal tangent line wherever f ′(x) = 0, an inflection point wherever f ″(x) changes sign, and is concave up wherever ...

Which of the following is the graph of y = x^ 3 + 3x^ 2 + 2?

Graph with one x-intercept at x = -1 and y-intercept at y = 2 (cubic shape).

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AP Calculus AB: 8.4.1 Graphs of Polynomial Functions
TermDefinition

Graphs of Polynomial Functions

• To graph a function:
1. Find critical points using the first derivative.
2. Determine where the function is increasing or decreasing.
3. Find inflection points using the second derivative.
4. Determine where the function is concave up or concave down.
• On an interval, the sign of the first derivative indicates whether the function is incr

note

  • Sketching an accurate drawing of a function requires several steps, even for a polynomial function.

  • First, you need to take the derivative of the function so that you can determine its critical points. Set the derivative equal to zero and solve for x. This function has two critical points.

  • Next, make a sign chart for the derivative. Choose a point from each interval to determine the sign of the derivative at that point. Then use arrows to indicate whether the function is increasing or decreasing. This function has a minimum at x = 7.5.

  • Next, take the second derivative of the function in order to check for possible inflection points. Set the second derivative equal to zero and solve for x.

  • Now make a sign chart to determine if the inflection point candidates are inflection points or not. Choose an x-value from each interval to determine the sign of the second derivative and the concavity of the function on that interval. Points where the function changes concavity are inflection points. This function has two inflection points.

  • Finally, use the information from the sign charts to sketch the graph of the function. The graph is decreasing for all points to the left of x = 7.5, and increasing to the right.

  • The graph is concave up to the left of x = 0 and to the right of x = 5. It is concave down in between. Therefore (0, 5) and (7.5, –1049.7) are points of inflection.

Use calculus to determine which of the following is the graph of f (x) = x^ 4 − 8x ^3 + 18x ^2 − 16x + 5.

Use your knowledge of relative extrema, critical points, concavity, points of inflection, and anything else (but a graphing calculator) to help you.

Use calculus to determine which of the following is the graph of
f(x)=2/3x^3+2x^2−30x.

Use your knowledge of relative extrema, critical points, concavity, points of inflection, and anything else (but a graphing calculator) to help you.

Which of the following graphs describes a function with these characteristics?
f ′(−2) = 0
f ′(2) = 0
f ″(−2) = 0
f ″(0) = 0
f ″(2) = 0
f ″(x) > 0 when x < −2
f ″(x) < 0 when −2 < x < 0
f ″(x) > 0 when 0 < x < 2
f ″(x) < 0 when x > 2

The graph of f (x) has a horizontal tangent line wherever f ′(x) = 0, an inflection point wherever f ″(x) changes sign, and is concave up wherever f ″(x) > 0 and concave down wherever f ″(x) < 0

Which of the following is the graph of y = x^ 3 + 3x^ 2 + 2?

Graph with one x-intercept at x = -1 and y-intercept at y = 2 (cubic shape).

Which of the curves has the following characteristics?
f ′(−2.8) = 0
f ″(−2) = 0
f ″(−0.25) = 0
for x < −2, f ″(x) > 0
for x > −0.25, f ″(x) > 0

This curve has minima or maxima ( f ′(x) = 0) at x = −2.8, −0.5 and 0. Its second derivative is 0 at x = −2 and x = −0.25 ( f ″ = 0) and these are the locations of inflection points. At values of x less than −2 and greater than −0.25, the second derivative is positive and the curve is concave up ( f ″(x) > 0).

Use your knowledge of calculus to determine which of the following is the graph of the function g (x) = x ^2 − 2x − 1.
Do not use a calculator.

The correct graph is the one with a vertex at (1, -2), opens upward, passes through (0, -1), and has x-intercepts at approximately -0.41 and 2.41.

Which of the following is the graph of y=x^3−3x−2?

The graph is a cubic curve with turning points at x=±1x = \pm 1x=±1 and roots at x=−2x = -2x=−2 and x=1x = 1x=1.