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Calculus and Vectors – How to get an A+ 2.2 Derivative of Polynomial Functions A Power Rule Consider the power function: y = x n , x, n ∈ R . Then: c) ( x 5 )' n −1 n Ex 1. For each case, differentiate. a) (1)' b) (x)' ( x )' = nx d n x = nx n −1 dx ⎛1⎞ d) ⎜ ⎟ ⎝x⎠ ' ⎛ 1 ⎞ e) ⎜ 7 ⎟ ⎝x ⎠ Some useful specific cases: (1)' = 0 f) ( x )' ( x)' = 1 g) (3 x )' ( x )' = 1 ' h) ( 4 x 3 )' 2 x i) ( x π )' B Constant Function Rule Let consider a constant function: f ( x) = c, c ∈ R . Then: (c)' = 0 d c=0 dx Ex 2. For each case, differentiate. a) (−2)' C Constant Multiple Rule Let consider g ( x) = cf ( x) . Then: Ex 3. Differentiate each expression: a) − 2x 3 b) ( π )' [cf ( x)]' = cf ' ( x) d d f ( x) [cf ( x)] = c dx dx (cf )' = cf ' b) c) −3 x2 2x Ex 4. Differentiate. a) f ( x) = 2 − 3 x + 4 x 2 − 3 x 7 D Sum and Difference Rules [ f ( x) ± g ( x)] = f ' ( x) ± g ' ( x) d d d [ f ( x) ± g ( x)] = f ( x) ± g ( x) dx dx dx ( f ± g )' = f '± g ' 2.2 Derivative of Polynomial Functions ©2010 Iulia & Teodoru Gugoiu - Page 1 of 4 Calculus and Vectors – How to get an A+ b) g ( x) = 1 2 4 − + x x2 x5 c) h( x) = x −3 x x Ex 5. Find the equation of the tangent line at the point E Tangent Line To find the equation of the tangent line at the point 1 P (1,0) to the graph of y = f ( x) = x 2 − . P(a. f (a)) : x 1. Find derivative function f ' ( x) . 2. Find the slope of the tangent line using: m = f ' (a) 3. Use the slope-point formula to get the equation of the tangent line: y − f ( a ) = m( x − a ) Ex 6. Find the equation of the tangent line of the slope m = 2 to the graph of y = f ( x) = 2 x . Graph the function and the tangent line. Ex 7. Find the points on the graph of y = f ( x) = 2 x 3 − 3x 2 + 1 where the tangent line is horizontal. 2.2 Derivative of Polynomial Functions ©2010 Iulia & Teodoru Gugoiu - Page 2 of 4 Calculus and Vectors – How to get an A+ Ex 8. Find the equation of the normal line to the curve F Normal Line 2 If mT is the slope of the tangent line, then slope of y = f ( x) = x + at P (1,3) . x the normal line m N is given by: 1 mN = − mT Ex 9. Analyze the differentiability of each function. b) y = f ( x) = x 2 / 3 a) y = f ( x) = 3 x c) y = f ( x) =| x − 3 | 2.2 Derivative of Polynomial Functions ©2010 Iulia & Teodoru Gugoiu - Page 3 of 4 Calculus and Vectors – How to get an A+ H Differentiability for piece-wise defined functions Let consider the piece-wise defined function: ⎧ f1 ( x), x < a ⎪ f ( x) = ⎨c, x = a ⎪ f ( x), x > a ⎩ 2 Ex 10. Analyze the differentiability of each function at x = 1 . ⎧⎪ x 2 , x ≤ 1 a) f ( x) = ⎨ ⎪⎩2 x, x > 1 The function f is differentiable at x = a if: (a) the function is continuous at x = a (b) f1 ' (a) = f 2 ' (a) (the slope of the tangent line for the left branch is equal to the slope of the tangent line for the right branch). ⎧⎪ x 2 , x ≤ 1 b) a) f ( x) = ⎨ ⎪⎩ x, x > 1 ⎧⎪ x 2 , x ≤ 1 c) a) f ( x) = ⎨ ⎪⎩2 x − 1, x > 1 Reading: Nelson Textbook, Pages 76-81 Homework: Nelson Textbook: Page 82 #2cdf, 3ce, 4aef, 5b, 6b, 7a, 8b, 9b, 11, 14, 17, 20, 25iii, 28 2.2 Derivative of Polynomial Functions ©2010 Iulia & Teodoru Gugoiu - Page 4 of 4