Math 151 Section 5.7 Antiderivatives Antiderivative A function F is called the antiderivative of f on an interval I if F !( x ) = f ( x ) for all x in I. Example: Is the function F ( x) = x ln x ! x an antiderivative of f ( x) = ln x ? Example: Find an antiderivative of f ( x) = 2x . General Antiderivative If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C, where C is an arbitrary constant. Antiderivative Formulas Function Particular Antiderivative Function c ! f ( x) f ( x) ± g ( x) x n , n ! "1 x!1 ekx cos x sin x sec 2 x sec x tan x 1 x 2 +1 Example: Find the most general antiderivative. A. f ( x) = 7x 4 + 3x 2 + 7 B. f ( x) = x + 3 x 5 + 34 1 1! x 2 1 x x 2 !1 Particular Antiderivative Math 151 Example: Find f ( x ) . A. f !( x) = e4 x + sec x tan x B. f !( x) = 3 1 4 1 + 3 + + 3x 4 x 5x x e C. f !( x) = "2(1+ x 2 ) "1 Example: Find f ( x ) if f !( x) = 3x 2 +15e3x + 4 and f (0) = 7 . Example: Find f ( x ) if f !!( x) = 20x 3 + 3sin x , f (0) = 2 , and f !(0) = 8 . Math 151 Example: A ball is thrown upward with a velocity of 50ft/sec from the edge of a 150-ft building. A. Find a formula that gives the height of the ball after x seconds. B. When does the ball reach its maximum height? C. How fast is the ball traveling when it hits the ground? Direction Field A direction field (or slope field) is a graphical representation of the slopes of a function at a variety of points on the xy-plane. Example: The graph shows the direction field given by f !( x) . Graph f ( x ) for f (0) = 2 and for f (0) = −4. Math 151 Antiderivatives of Vector Functions The vector function R (t ) = X (t ),Y (t ) is an antiderivative of r (t ) = x (t ), y (t ) if R !(t ) = r (t ) . Example: Find the most general antiderivative of r (t ) = 3t 2 + 2,sec 2 t . Example: A projectile is fired with an initial speed of 400m/s at an angle of elevation of 60°. A. Find the position vector of this projectile at time t. B. Find the range of the projectile.