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Chapter 3 | Derivatives
140. Find the point on the graph of f ( x ) = x 3 such that the tangent line at that point has an x intercept of 6. 141. Find the equation of the line passing through the point P (3, 3) and tangent to the graph of f ( x ) = 6 x −1 . 142. Determine all points on the graph of
f ( x ) = x 3 + x 2 − x −1 for which a. the tangent line is horizontal b. the tangent line has a slope of −1. 143. f (1) =5, f ′(1) =3 and f ″(1) =−6.
130. Let h ( x ) = f ( x )+ g ( x ). Find a. h ′(1), b. h ′(3), and c. h ′(4). 131. Let h ( x ) = f ( x ) g ( x ). Find a. h ′(1), b. h ′(3), and c. h ′(4).
Find a quadratic polynomial such that
144. A car driving along a freeway with traffic has traveled s ( t ) = t 3 −6 t 2 +9 t meters in t seconds. a. Determine the time in seconds when the velocity of the car is 0. b. Determine the acceleration of the car when the velocity is 0. 145. [T] A herring swimming along a straight line has traveled s ( t ) = t 2 t 2 +2 feet in t seconds. Determine the velocity of the herring when it has traveled 3 seconds. 146. The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function P ( t ) = 8 t +3 0.2 t 2 +1 , where t is measured in years. a. Determine the initial flounder population. b. Determine P ′(10) and briefly interpret the result. 147. [T] The concentration of antibiotic in the bloodstream t hours after being injected is given by the function C ( t ) = 2 t 2 + t t 3 +50 , where C is measured in milligrams per liter of blood. a. Find the rate of change of C ( t ). b. Determine the rate of change for t =8, 12, 24, and 36. c. Briefly describe what seems to be occurring as the number of hours increases. 148. A book publisher has a cost function given by C ( x ) = x 3 +2 x +3 x 2 , where x is the number of copies of a book in thousands and C is the cost, per book, measured in dollars. Evaluate C ′(2) and explain its meaning.
f ( x ) g ( x ) .
132. Let h ( x ) =
Find
a. h ′(1), b. h ′(3), and c. h ′(4).
For the following exercises, a. evaluate f ′( a ), and b. graph the function f ( x ) and the tangent line at x = a . 133. [T] f ( x ) =2 x 3 +3 x − x 2 , a =2
134. [T] f ( x ) = 1 x − x 2 , a =1 135. [T] f ( x ) = x 2 − x 12 +3 x +2, a =0
2/3 , a =−1
136. [T] f ( x ) = 1 x − x
137. Find the equation of the tangent line to the graph of f ( x ) =2 x 3 +4 x 2 −5 x −3 at x =−1. 138. Find the equation of the tangent line to the graph of f ( x ) = x 2 + 4 x −10 at x =8. 139. Find the equation of the tangent line to the graph of f ( x ) = (3 x − x 2 )(3− x − x 2 ) at x =1.
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