Consider the function f(x) = 2 – 8x² on the interval [-5, 6]. (A) Find the average or mean slope of the function on this interval, i.e. f(6) – f(-5) 6 - (-5) (B) By the Mean Value Theorem, we know there exists a c in the open interval (-5, 6) such that f' (c) is equal to this mean slope. F this problem, there is only one c that works. Find it.

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A: Given,

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Q: Let f(x)=(x-2)(x2+4x-7). Find all the points on this curve where the tangent line is horizontal.

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A:

Q: 1. Use the limit definition of derivative (no shortcuts!) to find the derivative of f(x) = 3x² + 2x.

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Q: Why does the Mean Value Theorem not apply to the function f(x)=x^(1/2) on the interval [-1,1]?

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Q: 3. Find all -values where the graph of f(x) = x³ + 3x² + x + 3 has a horizontal tangent line.

A: Solution is given below:

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A: This question is based on application of Derivative.

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: