# F x x 3 19 x 2 x 1 x 2 2 x 1 ii f x x 2 1 x 1 or x 1

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i. f ( x ) = x 3 - 19 x 2 + x + 1 x 2 - 2 x - 1 ii. f ( x ) = ( x 2 + 1 x < - 1 or x > 1 x - 1 x 1 iii. f ( x ) = ( x 3 - 7 x x < 0 0 x > 0 iv. f ( x ) = x + 7 v. f ( x ) = x 2 - 25 x + 5 2. Find a value of c for each of the following functions making the function continuous: i. f ( x ) = ( | x | - 1 x 2 x 2 + c x > 2 ii. f ( x ) = ( x - 1 x - 2 x ≤ - 1 cx 2 + 1 x > - 1

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The Derivative: Know the definition of the derivative and be able to use it to calculate derivatives Be able to find equations for tangent lines at points Know what it means to be differentiable Understand relationship between slopes of secants and tangents and average and in- stantaneous rates of change Exercises: 1. Consider the function f ( x ) = x 2 + x + 1. i. Give the equation of the secant line through the points (1 , f (1)) and (2 , f (2)) and give its slope. ii. Using the definition of the derivative, compute f 0 (1) iii. Using the definition of the derivative, compute f 0 ( x ) 2. For the following give equations for the lines tangent to the given point i. f ( x ) = 1 x + 1 , x = 1. ii. f ( x ) = x 2 - 1, x = - 2. iii. f ( x ) = x 3 + 1, x = 2 iv. f ( x ) = x - 1 + x , x = 2 3. Find the equation for the line perpendicular to f ( x ) = x 2 + 2 x at x = 1. 4. Use linear approximations of f ( x ) = x near 1 and 4 to estimate 3.
Challenge Question: Given a quadratic (degree 2) polynomial, there is no gaurantee that there are any real roots. For example, the polynomial x 2 + 1 has no real roots. Can the same be said for cubic (degree 3) polynomials?
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