MATH
Exam1_115_Solutions calc1

Exam1_115_Solutions calc1 - M ATH 115 F IRST M IDTERM...

• Notes
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2 1. (2 points each) For each of the following, circle all statements which MUST be true. (a) Let f be a non-decreasing differentiable function defined for all x . f ( x ) 0 for all x . f ′′ ( x ) 0 for all x . f ( x ) = 0 for some x . (b) Let f and g be continuous at x = - 1 , with f ( - 1) = 0 and g ( - 1) = 3 . f · g is continuous at x = - 1 . g f is continuous at x = - 1 . f g is continuous at x = - 1 . (c) Let f be differentiable at x = 2 , with f (2) = 17 . lim x 2 f ( x ) = 17 . lim h 0 f (2 + h ) - f (2) h = 17 . lim h 0 f (2 + h ) - f (2) h exists. (d) Let f be defined on [ a, b ] and differentiable on ( a, b ) , with f ( x ) < 0 for all x in ( a, b ) . If a < c < d < b , then f ( c ) > f ( d ) . f ′′ ( x ) > 0 for some x in ( a, b ) . f is continuous on ( a, b ) . (e) Let f be a twice-differentiable function that is concave-up on ( a, b ) , with f ( a ) = 4 and f ( b ) = 1 . For some x in ( a, b ) , f ( x ) = 2 . 5 . For all x in ( a, b ) , f ′′ ( x ) 0 . f ( a ) f ( b ) .
3 2. If you pluck a guitar string, a point P on the string vibrates. The motion of the point P is given by g ( t ) = A cos(220 π t ) , where g ( t ) is the displacement (in mm) of P from its position before the string was plucked, t is the number of seconds after the string was plucked, and A is a positive constant. (a) (6 points) Sketch a graph of g ( t ) , for 0 t 1 / 55 , on the axes below. Be sure to indicate A on your sketch. t g ( t ) 1 55 12 880 9 880 6 880 A (b) (3 points) Sketch tangent lines to your graph at t = 6 / 880 , t = 9 / 880 , and t = 12 / 880 . Use these to write the numbers g (6 / 880) , g (9 / 880) , and g (12 / 880) in order from least to great- est.

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• Spring '08
• BLAKELOCK
• Derivative, Continuous function, audience members

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