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Calculus

Differentiability

Excel in math and science

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Concept Quizzes

Approximate Rate of Change
Slope of a Curve
Tangent Line at a Point
Local Linear Approximation
Intermediate Value Theorem
Mean Value Theorem
Rolle's Theorem
Graphs of Derivatives

Challenge Quizzes

Level 2
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Characteristics of graphs of f and f'

Given \(f'(x) = x^2 + 9x + k,\) if \(f(x)\) is an increasing function for all values of \(x,\) what is the range of \(k?\)

\[\displaystyle{\frac{81}{4} \leq k}\] \[\displaystyle{\frac{77}{4} \leq k}\] \[\displaystyle{\frac{83}{4} \leq k}\] \[\displaystyle{\frac{79}{4} \leq k}\]

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by Brilliant Staff

Suppose \(f(x)\) is a function such that \(y=f'(x)\) is given by the curve in the above diagram. If \(y= f'(x) \) is tangent to the \(x\)-axis at \(x=a=7,\) which of the following statements is NOT correct about \(f(x)?\)

There is at most one point of intersection with the \(x\)-axis. f(x) is decreasing for all values of \(x.\) \(f(7)\) is a local maximum. If \(f(7)\) is \(0, \) \(f(0)\) is positive.

Excel in math and science

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Suppose \(f(x)\) is a differentiable function such that the graph of \(y=f'(x)\) is as shown in the above diagram. If \[a = 2, b = 3, c = 5, d= 7,\] and \(f'(e) = f'(d) \) for all \(e \geq d,\) which of following statements is correct?

In \([-2, 2],\) \(f(x)\) attains a maximum value at \(f(0)\). \(f(5)\) is a local maximum. There are four extreme values between \(x=-3\) and \(x=7.\) \(f(x)\) is a linear function for \( x > 7 .\)

Excel in math and science

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Suppose \(f(x)\) is a function such that the graph of \(y=f'(x)\) is a line as shown in the above diagram. If \(a=19,\) which of following statements is correct?

\(f(x)\) is an increasing function for \(x \le 19\). \(f(19)\) is a local maximum. \(f(x)\) is an increasing function for \(x \ge 19\) . \(f(19)=0.\)

Excel in math and science

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by Brilliant Staff

Suppose \(f(x)\) is a function such that \(f'(x) = K(x-a)(x-b) ,\) where \(K\) is a positive constant. If \(a=5\) and \(b=10,\) which of following statements is NOT correct?

\(f(x)\) is a cubic function. If \(f(5)= 0,\) there are two intersection points between \(f(x)\) and the \(x\)-axis. \(f(5)\) and \(f(10)\) are always \(0.\) \(f(x)\) is a decreasing function on \([5,10]\).

Excel in math and science

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by Brilliant Staff

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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

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Differentiability

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

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Our wiki is made for math and science

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Concept Quizzes

Challenge Quizzes

Characteristics of graphs of f and f'

Excel in math and science

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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

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Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations, examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Master advanced concepts through explanations,
examples, and problems from the community.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.