#1 Calculus – Hustle #1 Calculus – Hustle National Convention 2007 National Convention 2007
A particle moves according to A particle moves according to the equations: the equations: x( t) =4 t3 + 6 t 2 - 24 t - 24 x( t) =4 t3 + 6 t 2 - 24 t - 24 y( t) =2 t3 + 3 t 2 - 12 t + 12 y( t) =2 t3 + 3 t 2 - 12 t + 12 z( t) = t3 +6 t 2 + 12 t + 20 z( t) = t3 +6 t 2 + 12 t + 20 At what instant(s) of time [value(s) of t] At what instant(s) of time [value(s) of t] is the particle not moving? is the particle not moving? NOTE: Time can be negative. NOTE: Time can be negative. If no such instant of time, write none. If no such instant of time, write none.
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#1 Calculus – Hustle #1 Calculus – Hustle National Convention 2007 National Convention 2007
A particle moves according to A particle moves according to the equations: the equations: x( t) =4 t3 + 6 t 2 - 24 t - 24 x( t) =4 t3 + 6 t 2 - 24 t - 24 y( t) =2 t3 + 3 t 2 - 12 t + 12 y( t) =2 t3 + 3 t 2 - 12 t + 12 z( t) = t3 +6 t 2 + 12 t + 20 z( t) = t3 +6 t 2 + 12 t + 20 At what instant(s) of time [value(s) of t] At what instant(s) of time [value(s) of t] is the particle not moving? is the particle not moving? NOTE: Time can be negative. NOTE: Time can be negative. If no such instant of time, write none. If no such instant of time, write none.
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #2 Calculus – Hustle #2 Calculus – Hustle National Convention 2007 National Convention 2007
(ip )n (ip )n Evaluate: Evaluate: n=1 n! n=1 n!
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#2 Calculus – Hustle #2 Calculus – Hustle National Convention 2007 National Convention 2007
(ip )n (ip )n Evaluate: Evaluate: n=1 n! n=1 n!
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #3 Calculus – Hustle #3 Calculus – Hustle National Convention 2007 National Convention 2007
For the equation For the equation x3+ y 3 +3 x 2 y - 3 x 2 + 3 xy 2 = - 11, x3+ y 3 +3 x 2 y - 3 x 2 + 3 xy 2 = - 11, using implicit differentiation find using implicit differentiation find dy dx at the point (2,- 1) . dy dx at the point (2,- 1) .
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#3 Calculus – Hustle #3 Calculus – Hustle National Convention 2007 National Convention 2007
For the equation For the equation x3+ y 3 +3 x 2 y - 3 x 2 + 3 xy 2 = - 11, x3+ y 3 +3 x 2 y - 3 x 2 + 3 xy 2 = - 11, using implicit differentiation find using implicit differentiation find dy dx at the point (2,- 1) . dy dx at the point (2,- 1) .
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #4 Calculus – Hustle #4 Calculus – Hustle National Convention 2007 National Convention 2007
Let A be the number of infinite Let A be the number of infinite discontinuities and B be the number discontinuities and B be the number of removable discontinuities in the of removable discontinuities in the x4-2 x 2 + 1 x4-2 x 2 + 1 function f( x) = . function f( x) = . x3 - x x3 - x What is A- B ? What is A- B ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#4 Calculus – Hustle #4 Calculus – Hustle National Convention 2007 National Convention 2007
Let A be the number of infinite Let A be the number of infinite discontinuities and B be the number discontinuities and B be the number of removable discontinuities in the of removable discontinuities in the x4-2 x 2 + 1 x4-2 x 2 + 1 function f( x) = . function f( x) = . x3 - x x3 - x What is A- B ? What is A- B ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #5 Calculus – Hustle #5 Calculus – Hustle National Convention 2007 National Convention 2007
Given 24 m2 of cardboard, you are to Given 24 m2 of cardboard, you are to construct an open box (that is, a box construct an open box (that is, a box with no top). In cubic meters, what is with no top). In cubic meters, what is the largest volume this box can occupy? the largest volume this box can occupy?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#5 Calculus – Hustle #5 Calculus – Hustle National Convention 2007 National Convention 2007
Given 24 m2 of cardboard, you are to Given 24 m2 of cardboard, you are to construct an open box (that is, a box construct an open box (that is, a box with no top). In cubic meters, what is with no top). In cubic meters, what is the largest volume this box can occupy? the largest volume this box can occupy?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #6 Calculus – Hustle #6 Calculus – Hustle National Convention 2007 National Convention 2007
What is the tangent line approximation What is the tangent line approximation of 14 , using 16= 4? of 14 , using 16= 4 ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#6 Calculus – Hustle #6 Calculus – Hustle National Convention 2007 National Convention 2007
What is the tangent line approximation What is the tangent line approximation of 14 , using 16= 4? of 14 , using 16= 4 ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #7 Calculus – Hustle #7 Calculus – Hustle National Convention 2007 National Convention 2007
A circle’s radius is growing at a rate A circle’s radius is growing at a rate of 3 m s . At time t= 0 s the radius of 3 m s . At time t= 0 s the radius of the circle is 1 m . How fast is the of the circle is 1 m . How fast is the area of the circle growing (in m2 s ) area of the circle growing (in m2 s ) at time t= 3 s ? at time t= 3 s ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#7 Calculus – Hustle #7 Calculus – Hustle National Convention 2007 National Convention 2007 A circle’s radius is growing at a rate A circle’s radius is growing at a rate of 3 m s . At time t= 0 s the radius of 3 m s . At time t= 0 s the radius of the circle is 1 m . How fast is the of the circle is 1 m . How fast is the area of the circle growing (in m2 s ) 2 area of the circle growing (in m s ) at time t= 3 s ? at time t= 3 s ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #8 Calculus – Hustle National Convention 2007 National Convention 2007 1 骣 e2 ln x 2 x- 3 x - 1 1 琪 骣 e2 ln x 2 x- 3 x - 1 2 琪 琪 2 2 Let A= 琪 0 3 x . 琪 2 Let A= 0 3 x . 琪 3 琪 -2 ln( 4x) ( 2 x - 3) 琪 3 琪 琪 -2 ln( 4x) ( 2 x - 3) 桫 桫 Now replace every entry in A with its Now replace every entry in A with its derivative with respect to x, call this derivative with respect to x, call this new matrix B. What is the determinant new matrix B. What is the determinant of B? of B?
Answer : ______Answer : ______Round 1 2 3 4 5 Round 1 2 3 4 5
#8 Calculus – Hustle #8 Calculus – Hustle National Convention 2007 National Convention 2007 1 骣 e2 ln x 2 x- 3 x - 1 1 琪 骣 e2 ln x 2 x- 3 x - 1 2 琪 琪 2 2 Let A= 琪 0 3 x . 琪 2 Let A= 0 3 x . 琪 3 琪 -2 ln( 4x) ( 2 x - 3) 琪 3 琪 琪 -2 ln( 4x) ( 2 x - 3) 桫 桫 Now replace every entry in A with its Now replace every entry in A with its derivative with respect to x, call this derivative with respect to x, call this new matrix B. What is the determinant new matrix B. What is the determinant of B? of B?
Answer : ______Answer : ______Round 1 2 3 4 5 Round 1 2 3 4 5 #9 Calculus – Hustle #8 Calculus – Hustle National Convention 2007 Let f( x) = sin ( ix) , what is f(19) ( x) ? (19) Let f( x) = sin ( ix) , what is f( x) ? (NOTE: i is the imaginary unit) (NOTE: i is the imaginary unit)
Answer : ______Answer : ______Round 1 2 3 4 5 Round 1 2 3 4 5
#9 Calculus – Hustle #9 Calculus – Hustle National Convention 2007 National Convention 2007 Let f( x) = sin ( ix) , what is f(19) ( x) ? (19) Let f( x) = sin ( ix) , what is f( x) ? (NOTE: i is the imaginary unit) (NOTE: i is the imaginary unit)
Answer : ______Answer : ______Round 1 2 3 4 5 Round 1 2 3 4 5
#10 Calculus – Hustle #9 Calculus – Hustle National Convention 2007 National Convention 2007 A particle moves with acceleration A particle moves with acceleration 骣t 骣t a( t) = cos 琪 . The velocity of this a( t) = cos琪 . The velocity of this 桫2 桫2 particle is zero at time t = 0 . What is particle is zero at time t = 0 . What is the particle’s speed at time t = 3p ? the particle’s speed at time t = 3p ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#10 Calculus – Hustle #10 Calculus – Hustle National Convention 2007 National Convention 2007
A particle moves with acceleration A particle moves with acceleration 骣t 骣t a( t) = cos 琪 . The velocity of this a( t) = cos琪 . The velocity of this 桫2 桫2 particle is zero at time t = 0 . What is particle is zero at time t = 0 . What is the particle’s speed at time t = 3p ? the particle’s speed at time t = 3p ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#10 Calculus – Hustle #11 Calculus – Hustle National Convention 2007 National Convention 2007 h h 骣 3 骣 3 Evaluate: lim琪 1 + Evaluate: lim琪 1 + h 桫 h h 桫 h
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#11 Calculus – Hustle #11 Calculus – Hustle National Convention 2007 National Convention 2007
h h 骣 3 骣 3 Evaluate: lim琪 1 + Evaluate: lim琪 1 + h 桫 h h 桫 h
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#11 Calculus – Hustle #12 Calculus – Hustle National Convention 2007 National Convention 2007 6 2 6 2 Evaluate: 2 dx Evaluate: 2 dx 3 x -1 3 x -1
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#12 Calculus – Hustle #12 Calculus – Hustle National Convention 2007 National Convention 2007
6 2 6 2 Evaluate: 2 dx Evaluate: 2 dx 3 x -1 3 x -1
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#12 Calculus – Hustle National Convention 2007 #13 Calculus – Hustle #13 Calculus – Hustle National Convention 2007 National Convention 2007
Let f( x) = x2 + 1, Let f( x) = x2 + 1, g( x) = fⅱ( f( x)) f( x) dx g( x) = fⅱ( f( x)) f( x) dx and g (0) = 2 , what is g( x) ? and g (0) = 2 , what is g( x) ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#13 Calculus – Hustle #13 Calculus – Hustle National Convention 2007 National Convention 2007
Let f( x) = x2 + 1, Let f( x) = x2 + 1, g( x) = fⅱ( f( x)) f( x) dx g( x) = fⅱ( f( x)) f( x) dx and g (0) = 2 , what is g( x) ? and g (0) = 2 , what is g( x) ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #14 Calculus – Hustle #14 Calculus – Hustle National Convention 2007 National Convention 2007
For f( x) = e2x -10 e x + 12 x , what is For f( x) = e2x -10 e x + 12 x , what is the greatest value of x for which the the greatest value of x for which the tangent line is horizontal? tangent line is horizontal?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#14 Calculus – Hustle #14 Calculus – Hustle National Convention 2007 National Convention 2007
For f( x) = e2x -10 e x + 12 x , what is For f( x) = e2x -10 e x + 12 x , what is the greatest value of x for which the the greatest value of x for which the tangent line is horizontal? tangent line is horizontal?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #15 Calculus – Hustle #15 Calculus – Hustle National Convention 2007 National Convention 2007
On what interval(s) is On what interval(s) is x3 x3 f( x) = + x2 -3 x + 1 both concave f( x) = + x2 -3 x + 1 both concave 3 3 up and decreasing? Please, express up and decreasing? Please, express your answer in interval notation i.e. your answer in interval notation i.e. x稳( a, b) ( c , d ) x稳( a, b) ( c , d )
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#15 Calculus – Hustle #15 Calculus – Hustle National Convention 2007 National Convention 2007
On what interval(s) is On what interval(s) is x3 x3 f( x) = + x2 -3 x + 1 both concave f( x) = + x2 -3 x + 1 both concave 3 3 up and decreasing? Please, express up and decreasing? Please, express your answer in interval notation i.e. your answer in interval notation i.e. x稳( a, b) ( c , d ) x稳( a, b) ( c , d )
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #16 Calculus – Hustle #16 Calculus – Hustle National Convention 2007 National Convention 2007
What is the area between the curve What is the area between the curve y= sin ( x) and the x-axis, from y= sin ( x) and the x-axis, from x = -p to x = p ? x = -p to x = p ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#16 Calculus – Hustle #16 Calculus – Hustle National Convention 2007 National Convention 2007
What is the area between the curve What is the area between the curve y= sin ( x) and the x-axis, from y= sin ( x) and the x-axis, from x = -p to x = p ? x = -p to x = p ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #17 Calculus – Hustle #17 Calculus – Hustle National Convention 2007 National Convention 2007
sec2 x sec2 x Evaluate: dx Evaluate: dx 1+ tan2 x 1+ tan2 x
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#17 Calculus – Hustle #17 Calculus – Hustle National Convention 2007 National Convention 2007
sec2 x sec2 x Evaluate: dx Evaluate: dx 1+ tan2 x 1+ tan2 x
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #18 Calculus – Hustle #18 Calculus – Hustle National Convention 2007 National Convention 2007
What is the arc length of the curve What is the arc length of the curve defined by x( t) = 3 t2 , y( t) = 4 t2 defined by x( t) = 3 t2 , y( t) = 4 t 2 from t = 0 to t = 2 ? from t = 0 to t = 2 ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#18 Calculus – Hustle #18 Calculus – Hustle National Convention 2007 National Convention 2007
What is the arc length of the curve What is the arc length of the curve defined by x( t) = 3 t2 , y( t) = 4 t2 defined by x( t) = 3 t2 , y( t) = 4 t 2 from t = 0 to t = 2 ? from t = 0 to t = 2 ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #19 Calculus – Hustle #19 Calculus – Hustle National Convention 2007 National Convention 2007
2 2 Evaluate: xe- x dx Evaluate: xe- x dx 0 0
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#19 Calculus – Hustle #19 Calculus – Hustle National Convention 2007 National Convention 2007
2 2 Evaluate: xe- x dx Evaluate: xe- x dx 0 0
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #20 Calculus – Hustle #20 Calculus – Hustle National Convention 2007 National Convention 2007
What is the maximum value of the What is the maximum value of the function f( x) = - x3 +3 x + 1 for function f( x) = - x3 +3 x + 1 for x �[ 3,3] ? x �[ 3,3] ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#20 Calculus – Hustle #20 Calculus – Hustle National Convention 2007 National Convention 2007
What is the maximum value of the What is the maximum value of the function f( x) = - x3 +3 x + 1 for function f( x) = - x3 +3 x + 1 for x �[ 3,3] ? x �[ 3,3] ?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #21 Calculus – Hustle #21 Calculus – Hustle National Convention 2007 National Convention 2007
What is the first derivative of sin2(x 2 ) What is the first derivative of sin2(x 2 ) with respect to x? with respect to x?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#21 Calculus – Hustle #21 Calculus – Hustle MAQ National Convention 2007 MAQ National Convention 2007
What is the first derivative of sin2(x 2 ) What is the first derivative of sin2(x 2 ) with respect to x? with respect to x?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #22 Calculus – Hustle #22 Calculus – Hustle National Convention 2007 National Convention 2007
Let f( x) be a differentiable, invertible Let f( x) be a differentiable, invertible function, such that f (2) = 3. Let function, such that f (2) = 3. Let g( x) = f-1 ( x) . What is f (2) given g( x) = f-1 ( x) . What is f (2) given that g (2) = 7 and g (3) = 11? that g (2) = 7 and g (3) = 11?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#22 Calculus – Hustle #22 Calculus – Hustle National Convention 2007 National Convention 2007
Let f( x) be a differentiable, invertible Let f( x) be a differentiable, invertible function, such that f (2) = 3. Let function, such that f (2) = 3. Let g( x) = f-1 ( x) . What is f (2) given g( x) = f-1 ( x) . What is f (2) given that g (2) = 7 and g (3) = 11? that g (2) = 7 and g (3) = 11?
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #23 Calculus – Hustle #23 Calculus – Hustle National Convention 2007 National Convention 2007
(4+h)2 -( 4 - h) 2 (4+h)2 -( 4 - h) 2 Evaluate: lim Evaluate: lim h 0 2h h 0 2h
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#23 Calculus – Hustle #23 Calculus – Hustle National Convention 2007 National Convention 2007
(4+h)2 -( 4 - h) 2 (4+h)2 -( 4 - h) 2 Evaluate: lim Evaluate: lim h 0 2h h 0 2h
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #24 Calculus – Hustle #24 Calculus – Hustle National Convention 2007 National Convention 2007
sin( x) - cos( x) + 1 sin( x) - cos( x) + 1 Evaluate: lim Evaluate: lim x 0 x3-3 x 2 + 3 x x 0 x3-3 x 2 + 3 x
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5
#24 Calculus – Hustle #24 Calculus – Hustle National Convention 2007 National Convention 2007
sin( x) - cos( x) + 1 sin( x) - cos( x) + 1 Evaluate: lim Evaluate: lim x 0 x3-3 x 2 + 3 x x 0 x3-3 x 2 + 3 x
Answer : ______Answer : ______
Round 1 2 3 4 5 Round 1 2 3 4 5 #25 Calculus – Hustle National Convention 2007 National Convention 2007 What is f (1) if What is f (1) if 3 2 f( x) =( - x -1) ( x + 1) ( 2 x - 3) ? 3 2 f( x) =( - x -1) ( x + 1) ( 2 x - 3) ?
Answer : ______Answer : ______Round 1 2 3 4 5 Round 1 2 3 4 5
#25 Calculus – Hustle #25 Calculus – Hustle National Convention 2007 National Convention 2007 What is f (1) if What is f (1) if 3 2 f( x) =( - x -1) ( x + 1) ( 2 x - 3) ? 3 2 f( x) =( - x -1) ( x + 1) ( 2 x - 3) ?
Answer : ______Answer : ______Round 1 2 3 4 5 Round 1 2 3 4 5
#25 Calculus – Hustle