SL Calculus Practice Problems


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1 Alei  Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram below shows part of the graph of the function f : α A P 5 Q B The graph intercepts the ais at A(, ), B(5, ) and the origin, O. There is a minimum point at P and a maimum point at Q. (a) The function ma also be written in the form f : α ( a) ( b), where a < b. Write down the value of (i) a; (ii) b. Find (i) f (); (ii) the eact values of at which f '() = ; (iii) the value of the function at Q. (7) (c) (i) Find the equation of the tangent to the graph of f at O. (ii) This tangent cuts the graph of f at another point. Give the coordinate of this point. () (d) Determine the area of the shaded region. (Total 5 marks). A ball is dropped verticall from a great height. Its velocit v is given b v = 5 5e.t, t where v is in metres per second and t is in seconds. (a) Find the value of v when (i) t = ; (ii) t =. \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
2 (i) Find an epression for the acceleration, a, as a function of t. (ii) What is the value of a when t =? (c) (i) As t becomes large, what value does v approach? (ii) As t becomes large, what value does a approach? (iii) Eplain the relationship between the answers to parts (i) and (ii). Alei  Desert Academ (d) Let metres be the distance fallen after t seconds. (i) Show that = 5t + 5e.t + k, where k is a constant. (ii) Given that = when t =, find the value of k. (iii) Find the time required to fall 5 m, giving our answer correct to four significant figures. (7) (Total 5 marks). The graph of = + + has a maimum point between = and =. Find the coordinates of this maimum point In this question, s represents displacement in metres, and t represents time in seconds. (a) The velocit v m s d s of a moving bod ma be written as v = = at, where a is a constant. dt Given that s = when t =, find an epression for s in terms of a and t. Trains approaching a station start to slow down when the pass a signal which is m from the station. The velocit of Train t seconds after passing the signal is given b v = 5t. (i) Write down its velocit as it passes the signal. (ii) Show that it will stop before reaching the station. (c) Train slows down so that it stops at the station. Its velocit is given b v = d s = at, where a is a constant. dt (i) Find, in terms of a, the time taken to stop. (ii) Use our solutions to parts (a) and (c)(i) to find the value of a. \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of (Total 5 marks) 6. Consider the function h () = 5. (i) Find the equation of the tangent to the graph of h at the point where = a, (a ). Write the equation in the form = m + c. (ii) Show that this tangent intersects the ais at the point ( a, ). (Total 5 marks) 7. An aircraft lands on a runwa. Its velocit v m s at time t seconds after landing is given b the equation v = 5 + 5e.5t, where t. (a) Find the velocit of the aircraft (i) when it lands; (ii) when t =. () Write down an integral which represents the distance travelled in the first four seconds. (c) Calculate the distance travelled in the first four seconds. After four seconds, the aircraft slows down (decelerates) at a constant rate and comes to rest when t =. (d) Sketch a graph of velocit against time for t. Clearl label the aes and mark on the graph the point where t =.
3 Alei  Desert Academ (e) Find the constant rate at which the aircraft is slowing down (decelerating) between t = and t =. (f) Calculate the distance travelled b the aircraft between t = and t =. (Total 8 marks) 8. The diagram on the left shows the graph of = f (). On the right hand grid sketch the graph of = f (). 9. Consider the function f () = + e. (a) (i) Find f (). (ii) Eplain briefl how this shows that f () is a decreasing function for all values of (ie that f () alwas decreases in value as increases). Let P be the point on the graph of f where =. Find an epression in terms of e for (i) the coordinate of P; (ii) the gradient of the tangent to the curve at P. (c) Find the equation of the tangent to the curve at P, giving our answer in the form = a + b. (d) (i) Sketch the curve of f for. (ii) Draw the tangent at =. (iii) (iv) Shade the area enclosed b the curve, the tangent and the ais. Find this area.. Consider the function f given b f () = +,. ( ) A part of the graph of f is given at right. The graph has a vertical asmptote and a horizontal asmptote, as shown. (a) Write down the equation of the vertical asmptote. () f () =.9 f ( ) =.9 f () =.99 (i) Evaluate f ( ). (ii) Write down the equation of the horizontal asmptote. (c) 9 7 Show that f () =, ( ). (7) (Total marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
4 Alei  Desert Academ 7 8 The second derivative is given b f () =,. ( ) (d) Using values of f () and f () eplain wh a minimum must occur at =. (e) There is a point of infleion on the graph of f. Write down the coordinates of this point. (Total marks). A car starts b moving from a fied point A. Its velocit, v m s after t seconds is given b v = t + 5 5e t. Let d be the displacement from A when t =. (a) Write down an integral which represents d. Calculate the value of d. Working: (a)..... The displacement s metres of a car, t seconds after leaving a fied point A, is given b s = t.5t. (a) Calculate the velocit when t =. Calculate the value of t when the velocit is zero. (c) Calculate the displacement of the car from A when the velocit is zero. Working: (a).... (c)... The function f () is defined as f () = ( h) + k. The diagram at right shows part of the graph of f (). The maimum point on the curve is P (, ). (a) Write down the value of (i) h; (ii) k. Show that f () can be written as f () = () 8 (c) Find f (). The point Q lies on the curve and has coordinates (, ). A straight line L, through Q, is perpendicular to the tangent at Q. (d) (i) Calculate the gradient of L. (ii) Find the equation of L. (iii) The line L intersects the curve again at R. Find the coordinate of R. (8) (Total marks). Let = g () be a function of for 7. The graph of g has an infleion point at P, and a minimum point at M. Partial sketches of the curves of g and g are shown below P(, ) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
5 g ( ) g ( ) Alei  Desert Academ = g ( ) = g ( ) Use the above information to answer the following. (a) Write down the coordinate of P, and justif our answer. Write down the coordinate of M, and justif our answer. (c) Given that g () =, sketch the graph of g. On the sketch, mark the points P and M. 5. The velocit v m s of a moving bod at time t seconds is given b v = 5 t. (a) Find its acceleration in m s. The initial displacement s is metres. Find an epression for s in terms of t. 6. The function f is defined b f : α (a) Write down (i) f (); (ii) f (). 5 () (Total 8 marks) Let N be the normal to the curve at the point where the graph intercepts the ais. Show that the equation of N ma be written as = Let g : α (c) (i) Find the solutions of f () = g (). (ii) Hence find the coordinates of the other point of intersection of the normal and the curve. (d) Let R be the region enclosed between the curve and N. (i) Write down an epression for the area of R. (ii) Hence write down the area of R. 7. The diagram below shows the graphs of f () = + e, g () = +,.5. 6 (6) (Total 6 marks) f g (a) (i) Write down an epression for the vertical distance p between the graphs of f and g. (ii) Given that p has a maimum value for.5, find the value of at which this occurs. (6) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 5 of 8 p.5.5
6 The graph of = f () onl is shown in the diagram at right. When = a, = 5. Alei  Desert Academ 6 (i) Find f (). (ii) Hence show that a = ln. 8 (c) The region shaded in the diagram is rotated through 6 about the  ais. Write down an epression for the volume obtained. (Total marks) 5.5 a.5 8. The equation of a curve ma be written in the form = a( p)( q). The curve intersects the ais at A(, ) and B(, ). The curve of = f () is shown in the diagram at right. (a) (i) Write down the value of p and of q. (ii) Given that the point (6, 8) is on the curve, find the value of a. (iii) Write the equation of the curve in the form = a + b + c. (i) d Find. 6 (ii) A tangent is drawn to the curve at a point P. The gradient of this tangent is 7. Find the coordinates of P. (c) The line L passes through B(, ), and is perpendicular to the tangent to the curve at point B. (i) Find the equation of L. (ii) Find the coordinate of the point where L intersects the curve again. 9. The diagram shows a rectangular area ABCD enclosed on three sides b 6 m of fencing, and on the fourth b a wall AB. Find the width of the rectangle that gives its maimum area. A B 6 () (6) (Total 5 marks). A particle moves with a velocit v m s given b v = 5 t where t. (a) The displacement, s metres, is when t is. Find an epression for s in terms of t. (6) Find t when s reaches its maimum value. (c) The particle has a positive displacement for m t n. Find the value of m and the value of n. (Total marks). If f () = cos, and f π =, find f (). Working:... (Total marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 6 of
7 Alei  Desert Academ. (a) Sketch the graph of = π sin,, on millimetre square paper, using a scale of cm per unit on each ais. Label and number both aes and indicate clearl the approimate positions of the intercepts and the local maimum and minimum points. Find the solution of the equation π sin =, >. () (c) Find the indefinite integral ( π sin ) and hence, or otherwise, calculate the area of the region enclosed b the graph, the ais and the line =. () (Total marks). A curve with equation =f () passes through the point (, ). Its gradient function is f () = +. Find the equation of the curve.... (Total marks). Given that f () = ( + 5) find (a) f (); f ( ). 5. The diagram shows the graph of the function = +, <. Find the eact value of the area of the shaded region.... (Total marks) (Total marks) 6. In this question ou should note that radians are used throughout. (a) (i) Sketch the graph of = cos, for making clear the approimate positions of the positive intercept, the maimum point and the endpoints. (ii) Write down the approimate coordinates of the positive intercept, the maimum point and the endpoints. (7) Find the eact value of the positive intercept for. Let R be the region in the first quadrant enclosed b the graph and the ais. (c) (i) Shade R on our diagram. (ii) Write down an integral which represents the area of R. (d) Evaluate the integral in part (c)(ii), either b using a graphic displa calculator, or b using the following information. d ( sin + cos sin ) = cos. (Total 5 marks) 7. In this part of the question, radians are used throughout. = + \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 7 of
8 The function f is given b f () = (sin ) cos. The diagram shows part of the graph of = f (). The point A is a maimum point, the point B lies on the  ais, and the point C is a point of infleion. Alei  Desert Academ A C (a) Give the period of f. () O B From consideration of the graph of = f (), find to an accurac of one significant figure the range of f. () (c) (i) Find f (). (ii) Hence show that at the point A, cos = (iii) Find the eact maimum value.. (d) Find the eact value of the coordinate at the point B. (e) (i) Find f (). (ii) Find the area of the shaded region in the diagram. (f) Given that f () = 9(cos ) 7 cos, find the coordinate at the point C. 8. Let f () =. Given that f =, find f (). Working: (9) () () () (Total marks)... (Total marks) 9. Note: Radians are used throughout this question. (a) Draw the graph of = π + cos, 5, on millimetre square graph paper, using a scale of cm per unit. Make clear (i) the integer values of and on each ais; (ii) the approimate positions of the intercepts and the turning points. Without the use of a calculator, show that π is a solution of the equation π + cos =. (c) (d) (e) Find another solution of the equation π + cos = for 5, giving our answer to si significant figures. Let R be the region enclosed b the graph and the aes for π. Shade R on our diagram, and write down an integral which represents the area of R. Evaluate the integral in part (d) to an accurac of si significant figures. (If ou consider it necessar, d ou can make use of the result ( sin + cos ) = cos.) (Total 5 marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 8 of
9 . Find (a) sin ( + 7); Alei  Desert Academ d e.... (Total marks). The derivative of the function f is given b f () =.5 sin, for. + The graph of f passes through the point (, ). Find an epression for f ()..... Consider functions of the form = e k (a) Show that e k = k ( e k ). Let k =.5 (i) Sketch the graph of = e.5, for, indicating the coordinates of the intercept. (ii) Shade the region enclosed b this graph, the ais, ais and the line =. (iii) Find the area of this region. (c) (i) d Find in terms of k, where = e k. The point P(,.8) lies on the graph of the function = e k. (ii) Find the value of k in this case. (iii) Find the gradient of the tangent to the curve at P.. Let f () =. Find (a) f (); f ( ). (Total marks).... The diagram shows part of the curve = sin. The shaded region π is bounded b the curve and the lines = and =. π π Given that sin = and cos eact area of the shaded region. =, calculate the... π π \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM 9 of
10 5. The diagram below shows a sketch of the graph of the function = sin (e ) where, and is in radians. The graph cuts the ais at A, and the ais at C and D. It has a maimum point at B. (a) Find the coordinates of A. The coordinates of C ma be written as (ln k, ). Find the eact value of k. (c) (i) Write down the coordinate of B. (ii) d Find. Alei  Desert Academ B A C D (6) (d) (i) Write down the integral which represents the shaded area. (ii) Evaluate this integral. (e) (i) Cop the above diagram into our answer booklet. (There is no need to cop the shading.) On our diagram, sketch the graph of =. (ii) The two graphs intersect at the point P. Find the coordinate of P. (Total 8 marks) 6. Given that g ( ) =, deduce the value of (a) g ( ); ( g ( ) + ). 7. Consider the function f () = cos + sin. (a) (i) Show that f ( π ) =. (ii) Find in terms of π, the smallest positive value of which satisfies f () =. The diagram shows the graph of = e (cos + sin ),. The graph has a maimum turning point at C(a, b) and a point of infleion at D. d Find.... (c) Find the eact value of a and of b. () (d) Show that at D, = π e. (e) Find the area of the shaded region. (Total 7 marks) 6 D C(a, b) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
11 d 8. It is given that = + and that = when =. Find in terms of (a) Find ( + sin ( + )). The diagram shows part of the graph of the function f () = + sin ( + ). The area of the shaded region is given b a f ( ). Alei  Desert Academ Find the value of a..... (a) Consider the function f () = +. The diagram below is a sketch of part of the graph of = f (). 5 Cop and complete the sketch of f (). (i) Write down the intercepts and intercepts of f (). (ii) Write down the equations of the asmptotes of f (). () (c) (i) Find f (). (ii) There are no maimum or minimum points on the graph of f (). Use our epression for f () to eplain wh. The region enclosed b the graph of f (), the ais and the lines = and =, is labelled A, as shown at right. (d) (i) Find f (). (ii) Write down an epression that represents the area labelled A. (iii) Find the area of A. (7) (Total 6 marks). The derivative of the function f is given b f () = e +, <. The graph of = f () passes through the point (, ). Find an epression for f (). Working: A \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
12 Alei  Desert Academ. Let f be a function such that f ( ) = 8. (a) Deduce the value of (i) f ( ) ; + (ii) ( f ( ) ). d If f ( ) = 8, write down the value of c and of d. c (a) (i)... (ii)... c =..., d =.... Let h () = ( ) sin ( ) for 5 5. The curve of h () is shown at right. There is a minimum point at R and a maimum point at S. The curve intersects the ais at the points (a, ) (, ) (, ) and (b, ). (a) Find the eact value of (i) a; (ii) b. The regions between the curve and the ais are shaded for a as shown. (i) Write down an epression which represents the total area of the shaded regions. (ii) Calculate this total area. (c) (i) The coordinate of R is.. Find the coordinate of S. (ii) Hence or otherwise, find the range of values of k for which the equation ( ) sin ( ) = k has four distinct solutions.. Let f () =. + (a) Write down the equation of the horizontal asmptote of the graph of f. Find f (). 6 (c) The second derivative is given b f () =. (+ ) Let A be the point on the curve of f where the gradient of the tangent is a maimum. Find the coordinate of A. (d) Let R be the region under the graph of f, between = and = () ( a, ) 5 5 R S ( b, ) () (Total marks) R (), as shaded in the diagram at right Write down the definite integral which represents the area of R. (Total marks) \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
13 5. The function f is given b f () = sin (5 ). (a) Find f " (). Write down f ( ). 6. Let f () = ( + ) 5. Find (a) f (); Alei  Desert Academ f () The curve = f () passes through the point (, 6). d Given that = 5, find in terms of The table below shows some values of two functions, f and g, and of their derivatives f and g. f () 5 g () 5 f () g () 6 Calculate the following. (a) d (f () + g ()), when = ; ( g' ( ) + 6). k 9. Given = ln 7, find the value of k. 5. The graph of = sin from π is shown at right. The area of the shaded region is.85. Find the value of k.... \\.psf\home\documents\desert \SL \7Calculus\LP_SLCalculus.doc on /8/ at :9 AM of
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