ASSIGNMENT BETA COVER SHEET

Transcription

1 Question Done Backpack Ready for test ASSIGNMENT BETA COVER SHEET Name Teacher Topic Teacher/student comment Drill A indices Drill B tangents Drill C differentiation Drill D normals Drill E gradient Section A Quiz Speed time graph suvat Suvat vertical motion 4 Suvat vertical motion 5 Speed time graph 6 suvat 7 Speed time Section B 8 Tangents 9 Tangents 0 Tangents Tangents Tangents and normals differentiation 4 gradient 5 transformations 6 Curve sketching 7 differentiation 8 transformations 9 Curve sketching 0 tangent normal Normal intersection Tangent and normal 4 Check your work X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

2 a b g d e z h q i k l m n o p r s t u j c y w No human investigation can be called real science if it cannot be demonstrated mathematically Leonardo da Vinci Y Double Maths Assignment b (beta) This assignment is to give you practice in Differentiation and an Introduction to Mechanics. Drill Drill are the very basic techniques you need to solve maths problems. You need to practise these until you can do them quickly and easily. n n Section A: Convert these to the form y a or f ( ) then differentiate them using the correct notation () y= () y= () f()= Section B: Find the equation of the tangent to the curve at the point where = 5 () y () y () 0 y Section C: Convert these to the form 4 () y () y n m then differentiate y () y 4 Section D: Find the equation of the normal to the curve at the point where = 7 () y () y () y 4 Section E Find the point on the curve where the gradient is () y () y () y X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

3 V Section A Mechanics Mechanics Quiz write down your answers and then discuss in class A B C D E X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

4 F G H Mechanics questions. A car starts from rest and accelerates uniformly for 5 seconds until it reaches a speed of ms -. It then maintains this speed for 40 seconds before decelerating uniformly to rest in 0 seconds. Draw a speed time graph to model this motion and calculate the total distance travelled by the car (distance travelled equals area under speed-time graph). X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

5 . This is a suvat table. Only model a situation with a suvat table if acceleration is constant s u v a t s = displacement in metres, u = initial velocity in metres per second, v = final velocity in metres per second, a = acceleration in metres per second, t = time in seconds The suvat table can also be drawn horizontally: s u v a t The five suvat equations are given below. Each equation has one of the 5 items missing. Write out (many times) and memorise the suvat equations For each of the situations below, make and complete a suvat table. Put a question mark in the bo for whichever piece of information is required by the question. Make sure you ve used displacement not distance, and make sure you re consistently using the same direction (up or down, or left or right) as the positive direction. For each situation, select an equation from the 5 listed above. The easiest way to do this is to see what information is missing from your table (neither given nor required by the question) and choose the equation with this variable missing. Write down the equation, sub in the values from your suvat table and then rearrange to find the required information. Don t forget units on your final answer! (a) (b) (c) (d) A particle moves in a straight line. When t = 0 its velocity is m s. When t = 4 its velocity is m s. Find its acceleration, assumed to be constant. A car is approaching traffic lights at 5 m s when the driver applies the brake and comes to a stop in 45 m. Find the deceleration, assumed constant, and the time taken to stop. A particle has constant acceleration 6 m s whilst travelling in a straight line between points A and B. It passes A at m s and B at 5 m s. Calculate the distance AB. A man on the top of a tower of height 45m holds his arm over the side of the building and drops a stone vertically downwards. The stone takes.0s to reach the ground. Use this information to prove that the value of acceleration due to gravity is 9.8 to significant figures.. A particle is projected vertically upwards with speed 4.5m s -. Find the total time for which it is m or more above its point of projection. 4. A particle is projected vertically upwards from a point O with speed u m s -. Two seconds later it is still moving upwards and its speed is u m s-. Find a) the value of u, b) the time from the instant that the particle leaves O to the instant that it returns to O. X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

6 5. The diagram is a speed-time graph representing the motion of a cyclist along a straight road. At time t = 0 s, the cyclist is moving with speed u m s -. The speed is maintained until time t = 5 s, when she slows down with constant deceleration, coming to rest when t = s. The total distance she travels in s is 5 m. Find the value of u. 6. A car travelling on a straight road slows down with constant deceleration. The car passes a road sign with speed 40 km h - and a post bo with speed of 4 km h -. The distance between the road sign and the post bo is 40 m. Find, in m s -, the deceleration of the car. 7. A train starts from rest at station A and accelerates uniformly at m s - until it reaches a speed of 0 m s -. For the net T seconds the train maintains this constant speed. The train then retards uniformly at m s -,until it comes to rest at a station B. The distance between the stations is 6 km and the time taken from A to B is 5 minutes. a) Sketch a speed-time graph to illustrate this journey. b) Show that 40 T 00 c) Find the value of T and the value of. d) Calculate the distance the train travels at constant speed. e) Calculate the time taken from leaving A until reaching the point half-way between the stations. X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

7 G Section B Core X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

8 . In each of the following y is given as a function of. Find the derived dy function d : (a) y 5 (b) y (c) y 5 y (d) (e) y (f) y 4. Find the coordinates of all the points on the curve y with gradient. 5. The diagram shows part of the curve with equation y = f(). The curve cuts the - ais at the points A(, 0) and B(4, 0). The point C(, ) is a maimum point. Using a separate diagram for each, describe the transformations below and sketch the transformed curve, stating the new coordinates of A, B and C: (a) y f (b) y = f() (c) y = f() (d) y f 6. Sketch the following curves of y = f (), stating the equations of the asymptotes and the coordinates of any ais intercept: () f ( ) () f ( ) () f ( ) dy 7. Find when: d 5 (a) y (b) y (c) y 8. The following is the graph of a function y = f () with a turning point at A, A Using a separate diagram for each, sketch the transformed graph showing the new coordinates of A. 4 (a) y f (b) y f (c) y f y f 5 (d) X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

9 9. Sketch these curves, showing where they meet the aes and mark clearly any asymptotes: (a) y = ( + ) ( ) (b) y 6 6 (c) y 5 (d) 0. (a) Find the equation of the tangent to (b) At which points is the tangent to 4y 0. A curve has the equation y, 0. The point P on the curve has coordinate. (a) Show that the gradient of the curve at P is. f y f ( ) at the point = 4 ( ) parallel to the line 4 (b) (c) Find an equation for the normal to the curve at P, giving your answer in the form y = m + c. Find the coordinates of the point where the normal to the curve at P intersects the curve again. The normals to the curve y 7 4 at the origin (0,0) and the point 4 A(,0) meet at N. Show the coordinates of N are (, ) 5 5. (a) Find the equations of the tangent and normal to the curve y = 4/ at the point where =. (b) Show that the tangent does not intersect the curve again. (c) Show that the normal does intersect the curve again, and find the coordinates of the point of intersection. 4. Now check your work! Are you ready for the assignment test? Answers Any wrong answers? Check with another student then v.moore@bhasvic.ac.uk with the correct one also checked by another student, thank you 9 Drill A) A) A) B) y 0 B) 0 y 5 0 B) 7 y C) C) 9 C) D) y 0 D) y 4 0 D) 7y 6 0 E) (, ) D) (, ) (, D) (, ) (, ) 9 9) X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

10 Mechanics quiz discuss in class Section A (a) 750m (a), b), 6s, c).75m, d) 47.5m () hint : use s ut at to form the quadratic 4.9t 4.5t 0 and solve for t t =.8 s ( s.f.) (4a) u= 9 ( s.f.) (4b) 6 s (5) u = 8 (6) 0.65 m s - ( d.p.) (7a) (7c) = 0. (7d) km (7e) 5 s Section B (8a) (8b) 4 (8c) (9a) (9b) (9c) (9d) (0a) (0b) (0c) - 4 (0d) (0e) (0, 0) (a) (b) (c) (d) (0, 7). Do this question without answers provided (a) (b) (c) 0 6 (d) + (e) (f) (4) (, ) (, ) Check your graphs on after you have drawn them! (5) sketches X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05

11 (6) sketches (a) asymptotes y =, = 0. Intercept (- ½, 0 ) (b) asymptotes y = 0, =. Intercept (0, - / ) (c) asymptotes y = 0, = 0. No intercepts. 7 (a) 4 (b) 5 5 (c) 7 5 Check your graphs on after you have drawn them! (8) sketches (9) sketches (0a) 6y 4 0 (0b) (, ¼) (-, -¼ ) (b) 7 y (c) 6, (a) + y 4 = 0, y = 0 (c) (, ) X:\Maths\TEAM - Doubles & Furthers\A doubles - Assignments 05-6\05-6 assignments\dmy()beta5-6 Updated: 8/09/05