Calculus first semester exam information and practice problems

Transcription

1 Calculus first semester exam information and practice problems As I ve been promising for the past year, the first semester exam in this course encompasses all three semesters of Math SL thus far. It is like a half-length IB Math SL exam. If you need another copy of the handout, it is located at this URL: The entire Math SL syllabus can be found here: You got a copy of the formula booklet a year ago, and it is your responsibility to bring an unmarked copy to class on the day of the test. If you need a new one, you may print it from the internet, but not on my printer. That formula booklet can be found here:

2 One of the best things you can do to prepare for a high-stakes exam is to know what the instructions say ahead of time. Here they are. censored

3 For IB exams, you are required to write your work and answers in black or dark blue ink, although graphs and diagrams may be done in pencil. You should bring both a black or dark blue pen and a pencil to the test. Section A answers are written in lined boxes on the test paper, like this: Section B answers are written in answer booklets. You will get a fourpage answer booklet with each paper.

4 1 Algebra 1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation. Applications. Examples include compound interest and population growth. 1.2 Elementary treatment of exponents and logarithms. Laws of exponents; laws of logarithms. Change of base. 1.3 The binomial theorem: expansion of (a + b) n, n Î. Calculation of binomial coefficients using Pascal s triangle and **. ** should be found using both the formula and technology. 1. Consider the infinite geometric series (a) For this series, find the common ratio, giving your answer as a fraction in its simplest form. (b) Find the fifteenth term of this series. (c) Find the exact value of the sum of the infinite series. 2. In a geometric series, u 1 = ** and u 4 = *. (a) Find the value of r. (b) Find the smallest value of n for which S n > The first three terms of a geometric sequence are u 1 = 0.64, u 2 = 1.6, and u 3 = 4. (a) Find the value of r. (b) Find the value of S 6. (c) Find the least value of n such that S n >

5 4. Solve log 2 x + log 2 (x 2) = 3, for x > Let ln a = p, ln b = q. Write the following expressions in terms of p and q. (a) ln a 3 b (b) ln *** 6. (a) Given that log 3 x log 3 (x 5) = log 3 A, express A in terms of x. (b) Hence or otherwise, solve the equation log 3 x log 3 (x 5) = 1.

6 7. The mass M of a decaying substance is measured at one minute intervals. The points (t, ln M) are plotted for 0 t 10, where t is in minutes. The line of best fit is drawn. This is shown in the following diagram. (a) The correlation coefficient for this linear model is r = State two words that describe the linear correlation between ln M and t. (b) The equation of the line of best fit is ln M = 0.12t Given that M = a b t, find the value of b.

7 2 Functions and equations 2.1 Concept of function f : x aaa f (x). Domain, range; image (value). Composite functions. Identity function. Inverse function f The graph of a function; its equation y = f (x). Function graphing skills. Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range. Use of technology to graph a variety of functions, including ones not specifically mentioned. The graph of y = f 1 (x) as the reflection in the line y = x of the graph of y = f (x). Note the difference in the command terms draw and sketch. 2.3 Transformations of graphs. Translations: y = f (x) + b; y = f (x a). Reflections (in both axes): y = f (x); y = f ( x). Vertical stretch with scale factor p: y = pf (x). Stretch in the x-direction with scale factor * : y = f (qx). Composite transformations. Note: translation by the vector ** denotes horizontal shift of 3 units to the right and vertical shift of 2 down. 2.4 The quadratic function x a ax 2 + bx + c: its graph, y-intercept (0, c). Axis of symmetry. The form x aaa a(x p)(x q), x-intercepts (p, 0) and (q, 0). The form x a a(x h) 2 + k, vertex (h, k). Candidates are expected to be able to change from one form to another. 2.5 The reciprocal function x aaa *, x 0: its graph and self-inverse nature. The rational function ****** and its graph. Vertical and horizontal asymptotes. Diagrams should include all asymptotes and intercepts. 2.6 Exponential functions and their graphs: x aaa a x, a > 0, x aaa e x. Logarithmic functions and their graphs: *********, x > 0, x aaa ln x, x > 0. Relationships between these functions: a x = e x ln a ; ***********, ******, x > Solving equations, both graphically and analytically. Solutions may be referred to as roots of equations or zeros of functions. Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach. Solving ax 2 + bx + c = 0, a 0. The quadratic formula. The discriminant = b 2 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots. Solving exponential equations. 2.8 Applications of graphing skills and solving equations that relate to real-life situations.

8 8. The following diagram shows the graph of y = f (x), for 4 x 5. (a) Write down the value of f ( 3). (b) Write down the value of f 1 (1). (c) Find the domain of f 1. (d) On the grid above, sketch the graph of f Let f (x) = 3x 2 and g(x) = ***, for x 0. (a) Find f 1 (x). (b) Show that ****************. Let ***********, for x 0. The graph of h has a horizontal asymptote at y = 0. (c) Find the y-intercept of the graph of h. (d) Hence, sketch the graph of h. (e) For the graph of h 1, write down the x-intercept; (f) For the graph of h 1, write down the equation of the vertical asymptote. (g) Given that h 1 (a) = 3, find the value of a.

9 10. Let *************, for x > 0. (a) Show that f 1 (x) = 3 2x. (b) Write down the range of f 1. Let g(x) = log 3 x, for x > 0. (c) Find the value of **********, giving your answer as an integer. 11. Let **** ***********, for x q. The line x = 3 is a vertical asymptote to the graph of f. (a) Write down the value of q. The graph of f has a y-intercept at (0, 4). (b) Find the value of p. (c) Write down the equation of the horizontal asymptote of the graph of f.

10 12. Let f (x) = 4 tan 2 x 4 sin x, *******. (a) On the grid below, sketch the graph of y = f (x). (b) Solve the equation f (x) = The function f is defined by **************, for 3 < x < 3. (a) On the grid below, sketch the graph of f. (b) Write down the equation of each vertical asymptote. (c) Write down the range of the function f.

11 3 Circular functions and trigonometry 3.1 The circle: radian measure of angles; length of an arc; area of a sector. Radian measure may be expressed as exact multiples of π or as decimals. 3.2 Definition of cos θ and sin θ in terms of the unit circle. Definition of tan θ as ***. Exact values of trigonometric ratios of 0, **, **, and their multiples. The equation of a straight line through the origin is y = x tan θ. 3.3 The Pythagorean identity cos 2 θ + sin 2 θ = 1. Double angle identities for sine and cosine. Relationship between trigonometric ratios. 3.4 The circular functions sin x, cos x, and tan x: their domains and ranges; amplitude, their periodic nature; and their graphs. Composite functions of the form f (x) = a sin(b(x + c)) + d. Transformations. Applications. 3.5 Solving trigonometric equations in a finite interval, both graphically and analytically. Equations leading to quadratic equations in sin x, cos x, or tan x. 3.6 Solution of triangles. The cosine rule. The sine rule, including the ambiguous case. Area of a triangle, *****. Applications. Examples include navigation, problems in two and three dimensions, including angles of elevation and depression.

12 14.In triangle ABC, AC = 5, BC = 7, ** = 48, as shown in the diagram. Find ** giving your answer correct to the nearest degree. 15.In triangle PQR, PQ is 10 cm, QR is 8 cm and angle PQR is acute. The area of the triangle is 20 cm 2. Find the size of angle ****. 16.The following diagram shows triangle ABC. AB = 7 cm, BC = 9 cm and **** = 120. (a) Find AC. (b) Find ****. diagram not to scale

13 17.The following diagram shows a pentagon ABCDE, with AB = 9.2 cm, BC = 3.2 cm, BD = 7.1 cm, *** =110, *** = 52 and *** = 60. (a) Find AD. (b) Find DE. (c) The area of triangle BCD is 5.68 cm 2. Find ***. (d) Find AC. (e) Find the area of quadrilateral ABCD. 18. Two boats A and B start moving from the same point P. Boat A moves in a straight line at 20 km h 1 and boat B moves in a straight line at 32 km h 1. The angle between their paths is 70. Find the distance between the boats after 2.5 hours. 19. In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm. Calculate (a) the size of *** (b) the area of triangle PQR.

14 4 Vectors 4.1 Vectors as displacements in the plane and in three dimensions. Components of a vector; column representation; ************. Algebraic and geometric approaches to the following: the sum and difference of two vectors; the zero vector, the vector v; multiplication by a scalar, kv; parallel vectors; magnitude of a vector, *; unit vectors; base vectors i, j, and k; position vectors *****; **************. 4.2 The scalar product (dot product) of two vectors. Perpendicular vectors; parallel vectors. The angle between two vectors. For non-zero vectors, v w = 0 is equivalent to the vectors being perpendicular. For parallel vectors, w = kv, ********. 4.3 Vector equation of a line in two and three dimensions: r = a + tb. The angle between two lines. Relevance of a (position) and b (direction). Interpretation of t as time and b as velocity, with ** representing speed. 4.4 Distinguishing between coincident and parallel lines. Finding the point of intersection of two lines. Determining whether two lines intersect.

15 20. Let v = **** and w = ****, for k > 0. The angle between v and w is Find the value of k. 21. Find the cosine of the angle between the two vectors 3i + 4j + 5k and 4i 5j 3k. 22. A line L passes through points A( 2, 4, 3) and B( 1, 3, 1). (a) (i) Show that *********. (ii) Find ****. The following diagram shows the line L and the origin O. The point C also lies on L. Point C has position vector ****. (b) Show that y = 2. (c) (i) Find ********. (ii) Hence, write down the size of the angle between C and L. (d) Hence or otherwise, find the area of triangle OAB.

16 5 Statistics and probability 5.1 Concepts of population, sample, random sample, discrete and continuous data. Presentation of data: frequency distributions (tables); frequency histograms with equal class intervals; box-and-whisker plots; outliers. Grouped data: use of mid-interval values for calculations; interval width; upper and lower interval boundaries; modal class. Outliers are defined as more than 1.5 IQR from the nearest quartile. 5.2 Statistical measures and their interpretations. Central tendency: mean, median, mode. Quartiles, percentiles. Dispersion: range, interquartile range, variance, standard deviation. Effect of constant changes to the original data. Applications. 5.3 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles. 5.4 Linear correlation of bivariate data. Pearson s product-moment correlation coefficient r. Scatter diagrams; lines of best fit. Equation of the regression line of y on x. Use of the equation for prediction purposes. Mathematical and contextual interpretation. Validity of interpolation versus extrapolation. 5.5 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A is ******. The complementary events A and A' (not A). Use of Venn diagrams, tree diagrams and tables of outcomes. 5.6 Combined events, P(A B). The non-exclusivity of or. Mutually exclusive events: P(A B) = 0. Conditional probability; the definition **********. Independent events; the definition P(A B) = P(A) = P(A B'). Probabilities with and without replacement. 5.7 Concept of discrete random variables and their probability distributions. Expected values (mean), E(X) for discrete data. Applications, including games of chance. 5.8 Binomial distribution. Mean and variance of the binomial distribution. Conditions under which random variables have this distribution. 5.9 Normal distributions and curves. Standardization of normal variables (z-values, z-scores). Properties of the normal distribution.

17 23.Consider the following cumulative frequency table. (a) Find the value of p. (b) Find (i) the mean; (ii) the variance. 24. A data set has a mean of 20 and a standard deviation of 6. (a) Each value in the data set has 10 added to it. Write down the value of (i) the new mean; (ii) the new standard deviation. (b) Each value in the original data set is multiplied by 10. (i) Write down the value of the new mean. (ii) Find the value of the new variance.

18 25. The following table shows the average weights (y kg) for given heights (x cm) in a population of men. The relationship between the variables is modelled by the regression equation y = ax + b. (a) Write down the value of a and of b. (b) Hence, estimate the weight of a man whose height is 172 cm. (c) (d) Write down the correlation coefficient. State which two of the following describe the correlation between the variables. 26.There are nine books on a shelf. For each book, x is the number of pages, and y is the selling price in pounds ( ). Let r be the correlation coefficient. (a) Write down the possible minimum and maximum values of r. (b) Given that r = 0.95, which of the following diagrams best represents the data? (c) For the data in diagram D, which two of the following expressions describe the correlation between x and y? perfect, zero, linear, strong positive, strong negative, weak positive, weak negative

19 6 Calculus 6.1 Informal ideas of limit and convergence. Limit notation. Definition of derivative from first principles as ***************. Derivative interpreted as gradient (slope) function and as rate of change. Tangents and normals and their equations. Use of both forms of notation, ** and f '(x), for the first derivative. Identify intervals on which functions are increasing or decreasing. 6.2 Derivative of x n (n ), sin x, cos x, tan x, e x and ln x. Differentiation of a sum and a real multiple of these functions. The chain rule for composite functions. The product and quotient rules. The second derivative. Extension to higher derivatives. 6.3 Local maximum and minimum points. Testing for maximum or minimum using change of sign of first derivative and sign of second derivative. Use of the terms concave up and concave down. Points of inflexion with zero and non-zero gradients. At a point of inflexion, f "(x) = 0 and changes sign. f "(x) = 0 is not a sufficient condition for a point of inflexion. Graphical behaviour of functions, including the relationship between the graphs of f, f ', and f ". Optimization. Applications. 6.4 Indefinite integration as anti-differentiation. Indefinite integral of x n (n ), sin x, cos x, * and e x. The composites of any of these with the linear function ax + b. Integration by inspection, or substitution of the form *********. 6.5 Anti-differentiation with a boundary condition to determine the constant term. Definite integrals, both analytically and using technology. Areas under curves (between the curve and the x-axis). Areas between curves. Volumes of revolution about the x-axis. The values of some definite integrals can only be found using technology. Students are expected to first write a correct expression before calculating the area or volume. 6.6 Kinematic problems involving displacement s, velocity v, and acceleration a. Total distance travelled, ***.

20 27. Let f (x) = x 3 2x 4. The following diagram shows part of the curve of f. The curve crosses the x-axis at the point P. (a) Write down the x-coordinate of P. (b) Write down the gradient of the curve at P. (c) Find the equation of the normal to the curve at P, giving your equation in the form y = ax + b. 28.(a) Let f (x) = e 5x. Write down f (x). (b) Let g (x) = sin 2x. Write down g (x). (c) Let h (x) = e 5x sin 2x. Find h (x).

21 29. Let ** ***********, for x (a) Find f '(1). Consider another function g. Let R be a point on the graph of g. The x- coordinate of R is 1. The equation of the tangent to the graph at R is y = 3x + 6. (b) Write down g'(1). (c) Find g(1). Let h(x) = f (x) g(x). Find the equation of the tangent to the graph of h at the point where x = The following diagram shows the graph of **********. The points A, B, C, D and E lie on the graph of f. Two of these are points of inflexion. (a) Identify the two points of inflexion. (b) (i) Find f '(x). (ii) Show that f ''(x) = ***********. (c) Find the x-coordinate of each point of inflexion. (d) Use the second derivative to show that one of these points is a point of inflexion.

22 31.The diagram below shows the graph of ƒ(x) = x 2 e x for 0 x 6. There are points of inflexion at A and C and there is a maximum at B. (a) Using the product rule for differentiation, find ƒ (x). (b) Find the exact value of the y-coordinate of B. (c) The second derivative of ƒ is ƒ (x) = (x 2 4x + 2) e x. Use this result to find the exact value of the x-coordinate of C. 32.Let f (x) = cos 2x and g(x) = ln(3x 5). (a) Find f (x). (b) Find g (x). (c) Let h(x) = f (x) g(x). Find h (x).

23 33.Consider f (x) = ***x 3 + 2x 2 5x. Part of the graph of f is shown below. There is a maximum point at M, and a point of inflexion at N. (a) Find f (x). (b) Find the x-coordinate of M. (c) Find the x-coordinate of N. (d) The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form y = ax + b. 34. Let f '(x) = 24x 3 + 9x 2 + 3x + 1. (a) There are two points of inflexion on the graph of f. Write down the x-coordinates of these points. (b) Let g(x) = f ''(x). Explain why the graph of g has no points of inflexion.

24 35.Let y = f (x), for 0.5 x 6.5. The following diagram shows the graph of f, the derivative of f. The graph of f has a local maximum when x = 2, a local minimum when x = 4, and it crosses the x-axis at the point (5, 0). (a) Explain why the graph of f has a local minimum when x = 5. (b) Find the set of values of x for which the graph of f is concave down. 36. Let f '(x) = 6x 2 5. Given that f (2) = 3, find f (x). 37. Let f '(x) = 3x Given that f (2) = 5, find f (x).

25 38. Consider the functions f (x), g(x) and h(x). The following table gives some values associated with these functions. (a) Write down the value of g(3), of f (3), and of h (2). The following diagram shows parts of the graphs of h and h. There is a point of inflexion on the graph of h at P, when x = 3. (b) Explain why P is a point of inflexion. Given that h(x) = f (x) g(x), (c) find the y-coordinate of P. (d) find the equation of the normal to the graph of h at P.