3. Go over old quizzes (there are blank copies on my website try timing yourself!)

Transcription

1 final exam review General Information The time and location of the final exam are as follows: Date: Tuesday, June 12th Time: 10:15am-12:15pm Location: Straub 254 The exam will be cumulative; that is, it will cover all material from section 2.2 to 4.6. You ll need a scientific calculator to perform some of the computations. Graphing calculators will not be permitted. You may bring notes on the front and back of a half sheet of paper. The exam will have three sections, including around 7-10 true or false questions, 5-7 short answer questions, 7-10 free response questions, and possibly a bonus question. This review will not be collected for credit solutions will be posted by Sunday before the exam. Tips for studying I recommend the following strategy: 1. Start early 2. Understand every problem on this review 3. Go over old quizzes (there are blank copies on my website try timing yourself!) 4. Go over old exams (there are blank copies on Canvas try timing yourself!) 5. Go over old worksheets Topics Here are some key words to help you study. 1. Chapter 2: Limits and Derivatives The definition of a limit Limits that don t exist One-sided limits Limit laws Computing limits using algebra Computing limits using squeeze theorem The definition of continuity Types of discontinuities (removable, jump, and infinite) Intermediate value theorem (how to use this to show that an equation has a solution in a particular interval) 1

2 Limits involving infinity Computing derivatives by definition Finding equations of tangent lines Functions that are not differentiable (cusps, corners, and infinite tangent lines) Higher derivatives What f says about f (increasing/decreasing, concavity) 2. Chapter 3: Differentiation Rules Power, sum and constant multiple rules Product and quotient rules The derivative of e x Derivatives of trigonometric functions Chain rule Implicit differentiation Derivatives of inverse trig functions Derivatives of logarithmic functions Derivatives in the natural and social sciences Logarithmic differentiation Linear approximation 3. Chapter 4: Applications of Differentiation Related rates Local and global extrema Graph sketching Optimization Inderterminate forms L-Hospital s rule

3 Practice Problems True/False Questions: 1. True or False: The limit lim x 2 x 2 4 x 2 is undefined 2. True or False: A differentiable function must be continuous. 3. True or False: A continuous function is always differentiable. 4. True or False: If f(x) = 3 x then f (x) = x 3 x True or False: If f and g are differentiable functions, then (f/g) = f /g. 6. True of False: If x is a function of y, then d dy (x2 ) = 2x. 7. True of False: The function f(x) = x 2 + 2x + 1 has a global maximum value. 8. True of False: The function f(x) = x 2 + 2x + 1 has a global minimum value. 9. True of False: If the slope of the tangent line to the curve y = f(x) at x = 1 is 3, then the slope of the tangent line to the curve y = 2f(x) at x = 1 is True of False: If f (a) < 0, then f is decreasing at x = a. 11. If f(x) = 2 ln(3) then f (x) =

4 Computing Limits: 1. Compute each of the following limits, or state that it does not exist. If you use L Hospital s rule, state why you can do so. a) lim x 3 x 3 x 3 b) lim x sin(x) c) lim x 3 f(x), where f(x) = 1 d) lim x 0 x e) lim x 2 11 x 3 x x + x 2 f) lim x 2x g) lim x e x + 3 h) lim x e ln(x 3 ) + 1 x + x 2 i) lim x 1 2x 2 j) lim x 0 x sin(x) x tan(x) k) lim x (ex + x) 1/x l) lim x 0 sin(1/x) m) lim x x2 sin(1/x) { (x 3) 2 if x < 3, 9 3x if x > Use the definition of the derivative to find f (3) for f(x) = 2 x Show that the polynomial f(x) = x 4 x 2 1 has a root between x = 1 and x = 2.

5 Computing Derivatives: 4. Suppose that f(1) = 3, g(1) = 2, f (1) = 0, f (2) = 1, f(2) = 0, and g (1) = 5. Compute h (1) for the following. a) h(x) = f(x) g(x) f(x) 2 b) h(x) = f(x)g(x) c) h(x) = 3 (f g)(x) d) h(x) = f(x)x g(x) 5. Compute the following. a) f (1), given f(x) = 3x 2 2x + x 4 b) dy dx, given y = 2x + 3e x c) dz dw, given z = w3/2 (w + 10e w2 ) d) f (0), given f(x) = 1 x2 e x e) dy dt if yt = csc(2 3t) x + 4 x f) y, given y = (x + (x + sin 2 (x)) 2 ) 4 g) f (x), given f(x) = e 2 arctan x h) dp, given x 2 p 3 = x + p 2 dx (2,1) i) dx dy, given ex y y 2 = 10 j) f (t), given f(t) = arccos 1 t + t k) f (x), given f(x) = tan 3 (ln x + 1) l) dy dx, given y = (x2 + 1) ln(x2+1) x 3 x arcsin(x) m) dy dx, given sin(x + y) = yex + 2y Tangent Lines and Linear Approximation: 6. Find the equation of all horizontal tangent lines to y = 2e x 4x Find the equation of the tangent line to y = e x cos(x 2 ) at the point (0, 1).

6 8. Find the equation of the tangent line to x 2/3 + y 2/3 = 4 at the point ( 3 3, 1). 9. Use linear approximation to give a reasonable estimate of the following. Determine if each value is an overestimate or an underestimate. Provide reasoning for your answers. a) b) Critical Points and Extrema: 10. Find all critical points of h(x) = 3x 2 log(x). Determine whether each is a local maximum, minimum, or neither. 11. Find the absolute maximum and minimum value of f(x) = 3x 3/2 x 5/2 on [1, 3]. Graphing: 12. Sketch the graph of a function satisfying all of the following conditions: f (x) > 0 if 2 < x < 2, f (x) < 0 if x > 2 or x < 2, f (2) = 0, lim f(x) = 1, x f( x) = f(x), f (x) < 0 if 0 < x < 3, and f (x) > 0 if x > For each function, find its (i) domain, (ii) intercepts, (iii) asymptotes, (iv) intervals of increase/decrease, (v) local extrema, (vi) intervals of concave up/down, and (vii) points of inflection. Use this information to sketch a complete graph of the function. a) f(x) = x 2 e x/3 b) f(x) = ln(x )

7 Applications: 14. If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y = 10t 1.86t 2. a) Find the average velocity over the time interval [1, 2]. b) Find the instantaneous velocity when t = 1. Don t forget to include units! c) Find the acceleration when t = 1. Don t forget to include units! 15. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s(t) = 2 cos(t) + 3 sin(t), where s is measured in centimeters and t in seconds. Find the velocity and acceleration at time t. Interpret what you find in the context of the problem A population grows on an island according to the logistic model, P (t) =, where 1 + 4e 0.5t P is in thousands of people, and t is in years after the beginning of How quickly will the population be growing at the start of 2015? 17. A particle moves with position function s(t) = te t/2, where s is measured in meters and t in minutes. a) Find the velocity after t minutes. b) At what time(s) is the particle at rest? c) On what interval(s) is the particle moving the positive direction? d) Find the acceleration after t minutes. e) When is the particle speeding up? Slowing down? Related Rates: 18. A car starts 3 miles directly south of a library. It then drives due east at a speed of 10 mi/h. At what rate is the distance between the car and the library increasing when the car has driven for 5 miles? 19. A man walks along a straight path. A searchlight is located on the ground 15 feet from the path, and follows the man as he walks. When the man is 20 feet from the point on the path closest to the searchlight, he is walking at a speed of 5 feet per second. At this moment, at what rate is the searchlight rotating? 20. The volume of a cube is increasing at a rate of 10 cm 3 /min. How fast is the surface area increasing when the length of an edge is 30 cm? 21. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm 3 /sec, how fast is the water level rising when the water is 5 cm deep?

8 Optimization: 22. Currently, a small coffee shop sells about 40 cups of coffee per day, each for $ For every 10 cent increase in price, the average sales per day decreases by 2 cups. a) Find the demand function, p(x). b) What price should the shop sell coffee for if they want to maximize their revenue? 23. At what point(s) does the tangent line to y = x 3 3x 5 have the largest slope? 24. A city on the north side of a 1/5-mile-wide river is 5 miles down river from another city on the south side of a river. Say that we want to build a highway to connect these two cities. If it costs $8 million per mile to build on land and $20 million per mile to build across land, decribe the most economical way to build the highway. 25. A restaurant has a seating area that can seat up to 25 tables with 4 people per table. If they only put in 15 tables, they can charge an average $20 per meal and fill the restaurant. However, for each additional table they put in they must decreasing the price of the average meal by $1. Assuming all of the tables will be filled, what number of tables will maximize their profit? 26. A rectangle has one side on the x-axis, one side on the y-axis, one vertex at the origin, and one vertex on the curve y = e 2x, x 0. Find the maximum possible area of the rectangle. Find the minimum possible perimeter. 27. A rectangular storage container with an open top is to have a volume of 40 m 3. The lenth of its base is twice the width. Material for the base costs $ 15 per square meter. Material for the sides costs $ 4 per square meter. Find the cost of materials for the cheapest such container.