42S Calculus EXAM PREP University of Winnipeg June 5, Name:

Transcription

1 42S Calculus EXAM PREP University of Winnipeg June 5, 2015 Name:

2 The following topics in the James Stewart Single Variable Calculus textbook will be covered on the UW Final exam: Appendix A: Polynomials, Sign Diagrams, Inequalities, and Absolute Values 2.2 The limit of a function 2.3 Calculating limits using the limit laws 2.5 Continuity 3.5 Trigonometric limits 3.1 Derivatives and rates of change 3.2 The Derivative as a function 3.3 Differentiation formulas 3.4 Derivatives of trigonometric functions 3.5 The chain rule 3.6 Implicit differentiation 7.2 Derivatives of Natural Logarithmic Function 7.3 Derivatives of Natural Exponential Function 7.4 Derivatives of General Logarithmic and Exponential Functions 7.6 Derivatives of Inverse Trigonometric Functions 3.8 Related Rates 4.1 Maximum and minimum values 4.2 The Mean Value Theorem 4.3 How Derivatives Affect the Shape of a Graph 4.4 Limits at infinity; horizontal asymptotes 4.5 Summary of Curve sketching 4.7 Optimization problems 4.9 Antiderivatives Appendix E: Sigma Notation 5.1 Area and Distance 5.2 The Definite Integral 5.3 The Fundamental Theorem of Calculus 5.4 Indefinite integrals and the Net Change Theorem 5.5 The Substitution Rule 6.1 Areas between curves 6.2 Volumes 6.5 The Average Value of a Function 7.2 Integrals of Natural Logarithmic Function 7.3 Integrals of Natural Exponential Function 7.4 Integrals of General Logarithmic and Exponential Functions 7.6 Integrals of Inverse Trigonometric Functions 8.1 Integration by Parts 8.2 Trigonometric Integrals 8.3 Trigonometric Substitution 8.4 Integration of Rational Functions by Partial Fractions 8.5 Strategy for integration 7.8 Indeterminate forms and L Hospital s Rule 8.8 Improper integrals 9.1 Arc Length

3 MATH-1101/1103/1104 Introduction to Calculus March 2010 The following is a list of Theorems that you must know for the Final Examinations. 1. Use the definition of the derivative to prove: (a) The Product Rule, i.e., (fg) = f g + fg. (b) The Quotient Rule, i.e., ( f g ) = f g fg. g 2 d (c) sin x = cos x. (d) dx d dx cos x = sin x. 2. Must be able to prove: If f is differentiable at x, then f is continuous at x. 3. Must be able to define sin 1 x, cos 1 x tan 1 x and prove the following: (a) (b) (c) d dx sin 1 x = 1 1 x 2 for x ( 1, 1) d dx cos 1 (x) = 1 1 x 2 for x ( 1, 1) d dx tan 1 x = 1 1+x 2 for x (, ) 4. Must be able to state Rolle s Theorem and the Mean Value Theorem(for derivatives) and be able to use Rolle s Theorem to prove the Mean Value Theorem. 5. Must be able to show that if f is differentiable on an interval (a, b) and (a) If f (x) > 0 for all x (a, b), then f increases on (a, b). (b) If f (x) < 0 for all x (a, b), then f decreases on (a, b). (c) If f (x) = 0 for all x (a, b), then f is constant on (a, b). 6. Must be able to state and prove Fermat s theorem. 7. Must be able to state both the first and second part of the Fundamental Theorem of Calculus and be able to prove the second part of the Fundamental Theorem of Calculus using the first part. 8. Must be able to state and prove the Mean Value Theorem for integrals. 9. Must be able to define the natural logarithmic function and be able to show that ln(pq) = ln(p) + ln(q) for all reals p, q > Must be able to define the exponential function and be able to prove d dx exp(x) = exp(x). 11. Must be able to show that if u = f(x) and v = g(x) are differentiable functions of x, then f(x)g (x) dx = f(x)g(x) g(x)f (x) dx, i.e., u dv = uv v du. 12. Must be able to apply ANY theorems developed in the course.

4

5

6

7

8 MATH-liOl (6) Special Instructions: Intro to Calculus Final Examination There are sixteen questions. Answer all of the questions in the booklet provided. Use the backs of pages for rough work if necessary. Calculators or similar electronic aids are NOT permitted. Return the exam with your booklet. Value Differentiate the following: [Do not simplify] (a)f(x)=csc2x+y -+ir (b) f(x) = (sin 1(3x) + 4x2) (c) f(x) = tan 3 (e_sx + x1/3) (d) f(x) (2 og 5\sece z sec 1(x) ±tan (x 2) cotx) (e) f(x) = (x2 + 1)c dt (f)f(x)=j 5 ln(sinx) t Find the equation of the tangent line to the curvex2y2 + ln(xy) + y = 2 at the point (1,1) Evaluate each of the following integrals: 1 (a) I (e_2x + + J \ ijl x 2 (b) fxtan_1(x)dx f (c) sec 6 x tan 4 x dx 1 + sec 5x + dx csc3x j (d)f32 dx (e)i J I 7x 2 + 4x + 2 4x 3+x dx

9 (c) hm MATH-i 101(6) Page 2 of 3 Value 3 4. (a) Find the area bounded by the curves y = cos x, and y sin(2x) from x = 0 to x = ir/2. 3 (b) Find the volume of the solid obtained by rotating about the y-axis the region bounded by the cuvers y =lnx,y l,y= 2,x= Given that f(x) - 2 f (x) = 2x2(x f (x) 2 6x± 12) 2x(x (a) Construct sign diagrams for f (x) and f (x). Identify the intervals where f(x) is increasing/decreasing and concave up/concave down, using interval notation. (b) Sketch the graph of f. Label all intercepts, local extreme values, all points of inflection, and any horizontal and vertical asymptotes A 13-foot ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec. At what rate is the angle 6 between the ladder and the ground changing then? Anopen-top box is to be made by cutting equal squares from the corners of a 12 inch by 12 inch sheet of tin and bending up the sides. Find the dimensions of the box so that it has maximum volume Evaluate the limits. Do NOT use l Hospital s Rule. (a) x2 3x 4 hm - i x3 + 1 (b) lim(x2(3x)) 2cot x-+o 6x+1 \/9x + 8x Evaluate the limit. [You may use l Hospital s Rule.] (a)lim x-+o (b) lim(1 e X S1flX 2x) 2x Using the definition of definite integral, evaluate I (x2 J 1 x) dx

10 MATH-i 101 (6) Page 3 of 3 Value (a) State what it means for a function 3 (b) Prove that if f is differentiable at a, then f to be continuous at a. f is continuous at a State the Mean Value Theorem for Derivatives Prove that if increases on (a, b). f is differentiable on interval (a, b) and if f (x) > 0 for all x e (a, b), then f State both parts of the Fundamental Theorem of Calculus (FTC 1 and FTC 2) Determine whether the following improper integral is convergent or divergent. If it is convergent, evaluate the integral. f=dx Find the arc length of the curve y = lnsinx from x = 7r/4 to x = ir/3. Total 100

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33 4 4. Find the function that satisfies,

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62