Show all your work. If you use the calculator, say so and explain what you did. f(x) =(2x +5) 1 3

Transcription

1 Old Exams, Math 142, Calculus, Dr. Bart Show all your work. If you use the calculator, say so and explain what you did. 1. Find the domain and range of the following functions: f(x) = p x 2 ; 4 f(x) =ln(x 2 ; 25) f(x) =(2x +5) Suppose that f is an even function, and that ;12 f(x) 9whenx 0. a. Is it possible that f(;15) = ;15? Explain. b. Is it possible that f(;15) = ;3? Explain. c. Suppose that f(x) is concave up and decreasing for x 2. What do we know about f(x) for x ;2? Explain. d. Sketch a possible graph of f(x). 3. To nd the age of objects, scientists often use a technique called carbon dating. The amount of radio active carbon decreases exponentially. We nd that the amount of carbon (C) after t years is given by : C = C 0 e ;:000124t where C 0 is the original amount of radio active carbon. Suppose we are given two paintings both of which are claimed to be 15th century masters. a. Painting number 1 was tested and the amount of carbon is 97:5% of the original amount of radioactive carbon. (i.e.c = :975C 0 )How old is this painting? Could it be authentic? b. Painting number 2 was tested and the amount of carbon is 93:4% of the original amount of radioactive carbon. How old is this painting? Could it be authentic? 4. Suppose that f(x) is obtained from y =cos(x) via the following transformations: shrink horizontally by a factor of 3, stretch vertically by a factor of 2, followed by a shift up by 1. a. Write down a formula for f(x). b. What are the domain and range of f? 5. Find two functions f(x) and g(x), so that h(x) =f g(x) for the following functions: 1

2 h(x) = p 1 ; x 2 h(x) = tan(x 3 +7) x h(x) =ln x +1 h(x) =e x2 6. Giventhegraphoffasshown in for example in prblm 2 pg 127, a. Where is f 0 (x) =0? b. Estimate f 0 (1), and f 0 (3). c. On the same set of axes sketch a graph of f'. 7. Given the graph of g'(x) in for example prblm 4 pg127. a. Where does g have stationary points? b. Where does g have local maxima? Local minima? Explain. 8. Given the following graph of f (see for instance prblm 2-27 in 2.7), evaluate the following limits: x!0 x!;3 x!;1 lim x!2; f(x) x!2 + x!2 x!0 lim x!4 ; f(x) 9. Use your calculator to nd the following limit: sin(h) lim h!0 h 2

3 10. Use the limit denition of the derivative to compute f 0 (4) for f(x) =x Find the anti-derivatives of the following functions: p f(x) =2x x 2 ; 4 f(x) = 2x x 2 ; 25 f(x) = xex ; e x x 2 2 f(x) = 1+4x 2 f(x) =2x cos(x) ; x 2 sin(x) 12. a. Graph the equation f(x) = x2 +:01 on your standard screen. It looks like straight x line. Clearly f is NOT the equation of a line! (It does not look like y = mx + b!) Clearly the calculator does not give us the real picture, there must be some hidden behaviour. a. What is the domain of f? Most importantly, are there any vertical asymptotes? b. Take the derivative of f, and nd the intervals on which f is increasing. Then nd the intervals on which f is decreasing. c. Take the second derivative of f. Where is f concave up, and where is f concave down? d. Sketch a correct graph of f based on your work in a, b and c. Indicate the scale on your x- and y-axis. 13. Use your knowledge of calculus to nd the maximum and minimum values of f(x) =x 3 ; 3x + 5 on the interval [-2,2]. Show allyour work. 14. Find the equation for the tangentline to the function f(x) =e x + 1 at the point x = 1. (As always, show your work.) 15. a. Use implicit dierentiation to nd dy for the relation x2 +3xy + y 2 = 1. Show dx work. b. What is the slope of this curve at(x y) =(1 0)? Show work. 16. a. Use implicit dierentiation to nd dy for the relation 9x2 +4y 2 = 36. Show dx work. b. Where is the tangent line horizontal? Show work. c. Where is the tangent line vertical? Show work. 17. Use geometric methods to nd the following integral R 3 0 (2x+1)dx. Give asketch of the appropriate area and show all your work. 3

4 18. Suppose that the graph of f is as is given below. a. Suppose that g(x) = R x f(x)dx and that h(x) =R x 0 2 f(x)dx. Is g(x) h(x) oris g(x) h(x)? Explain your answer. b. Compute g(7) and h(7). Show your work. 19. Suppose that the population of a town is given by the equation P (t) =50 000(:96) t, where t is the number of years since Jan a. What was the population on Jan ? Show work. b. Use calculus to nd the rate of change of the population on Jan Show work. Is the population increasing or decreasing? 20. Let f(x) =xe x and let C(x) =ax 3 + bx 2 + cx + d a. Find f 0 (x) f 00 (x) f 000 (x) C 0 (x) C 00 (x) andc 000 (x). b. Set f(0) = C(0) f 0 (0) = C 0 (0) f 00 (0) = C 00 (0) and f 000 (0) = C 000 (0) to solve for a, b, c, and d. Show work. c. So, f(x)? 21. Suppose that f is decreasing, concave down, f(2) = 3, and f 0 (5) = ;4. a) Sketch a possible graph of f. b) How many zeros can f have? Why? c) What can you say about the location of the zeros. d) What is lim x!1 f(x)? e) Is it possible for f 0 (1) = ;1? f) Is it possible that f 0 (6) = ;1? 22. a) What is the denition of the derivative? b) Find the equation of the line that is tangent tof(x) = 1 at x =1=3. x;1 c) Graph f(x) and this tangent linetof(x) atx =1=3 labeling the x and y intercepts on both curves. 23. The population P, of China, in billions can be approximated by the function P =1:15(1:014) t where t is the number of years since the start of According to this model, how much did the population increase in 1993? How fast is the population growing at the start of 1993? How fast is the population growing at the start of 1994? Give your answers in millions of people per year. Explain the relationship between the three answers. 24. Let F (x) =e x2 (i) Find F 0 (x). Find R 2 1 2xex2 dx two way: a) Using left and right hand sums accurate to one decimal place. Say how many subdivisions you used. 4

5 b) Using the fundamental theorem of calculus. 25. Use implicit dierentiation to nd the equation of the tangent line to the curve p x + p y =7atx = a) What is the denition of the derivative? b) Calculate the derivative off(x) =5x 2 +3x using the denition. c) Calculate this derivative using your rules for dierentiation. 27. A y is crawling along a curved barrier on the xy-plane whose shape is given by y =7; x 2 : A spider waits at the point (0,4). Find the distance between the two creatures when they rst see one another. (Hint: Draw a picture.) 28. In any given locality, the length of daylightvaries during the year. In Des Moines, Iowa, the number of minutes of daylight onaday t days after the beginning of the year is given approximately by the formula D(t) = sin[ t;79:5 ] 0 t 160: 183 (I'll make thenumbers nicer on the exam.) a. Find the maximum and the minimum lengths of daylight and the dates at which they occur. b. Find an equation for the rate of change of daylight per day. c. What is the change in the amount ofdaylight on the 183 day of the year? d. When is the rate of change of daylight the largest? 29. Use implicit dierentiation to show that d dx ;1 (arccos x) = p 1;x If f(x) =1=3x 3 ; x 2 +3x + 1, use the rst and second derivative to nd the intervals on which f is increasing? decreasing? concave up? and concave down? 31. A cipheid variable is a star whose brightness alternately increases and decreases. Sara's Star is one such star. For this star the interval between maximum brightness is 6 days. The average brightness of Sara's Star is 5.0 and its brightness changes by 1=3. In view of this data: There is a trigonometric function modeling, B(t) =5+1=3 cos( t), the brightness t days after one of the times that Sara's 3 star is at it's brightest. a. Graph B(t). b. Find an equation for the rate of change of brightness after t days. c. Find the rate of decrease of Sara's Star's brightness after 2 days, to one decimal place. d. When is the brightness of Sara's Star increasing the most rapidly in the rst six days? 5

6 32. Suppose a yam at room temperature, 40 o C, is placed in an oven at temperature, 200 o C. Then the temperature of the yam t minutes after it is placed in the oven is given by f(t) =200; 160e ;0:023t. a) What are the units of f 0 (t)? b) Is f 0 (t) positive or negative? Why? c) What is the temperature of the yam twenty minutes after it is put in the oven? d) How much did the temperature of the yam increase in the twentieth minute? e) Find f 0 (10). f) Find lim t!1 f(t). Explain your answer. You may use a graphing calculator to substantiate your result. 33. Consider the region A bounded above by the graph of f(x) = e ;x2, bounded below by the graph of g(x) =e x2 ; 1, and bounded on the left by the y-axis. a. Sketch and label the curves f(x) andg(x) as accurately as possible and shade the region A. Find (approximately if necessary) and lable the coordinates of the three corner points of A. b. By just looking at your sketch in part (a), decide whether the area A is more or less that 0.7. Is it more or less than 0.3? Give a graphical justication for your answers. (Hint: Compare A to a convenient triangle or square.) c. Express the area of region A as an integral or the sum or dierence of integrals. 34. Dierentiate the following functions. Please do not simplify. No explinations needed. a) f(x) =e 3x sin x b) g(x) = ln(2xe x ) c) H(x) = arcsin(x + p x) d) K(x) =x x 2 ;3 35. Find the equation of the tangent line to the function, y 2 = x 3 (4 ; x) at(2 4). 36. a. What is the denition of the derivative? b. Explain the meaning of the denition of the derivativeinwords and with a picture. c. If f(x) =2x 2 +4x, use the denition of the derivative to show that f 0 (;2) = ;4 d. Find the equation of the tangent lineto2x 2 +4x at x = ;2. e. Graph f(x) =2x 2 +4x and the tangent to the curve atx = ;2. 6