Midterm 1 Review Problems Business Calculus

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1 Midterm 1 Review Problems Business Calculus 1. (a) Show that the functions f and g are inverses of each other by showing that f g(x) = g f(x) given that (b) Sketch the functions and the line y = x f(x) = x g(x) = x 3 2. Start with y = f(x) = x 2 and then through translations derive the sketch of the function y = f(x) = 1 2 (x 3) x 1 3. Find the domain of the function y = f(x) = (x 2 4) x 2 the number line or express it in the interval notation. 4. (a) Find limit lim x 1 x 1 x 1 (b) lim 3 x 2 (x 2 + 2x) x (c) lim x (x + 3) (d) Find limit lim x 0 x x (e) Find limit lim x 1 2 x 1 (f) lim x 3 x 2 x 12 x In the following problem use the definition of derivative: and label the domain on dy dx = lim f(x + h) f(x) h 0 h (a) Show that the derivative of any constant function f(x) = c is zero using the definition of derivative (b) Given that y = f(x) = x, find dy using the definition dx (c) Given that y = f(x) = cx, where c is a constant, find derivative using the definition (d) Given that y = f(x) = x 2 find derivative using the definition (e) Given that f(x) = x 3 find derivative using the definition (f) Given that f(x) = x 2 + x find derivative using the definition 1

2 (g) Given that f(x) = 1 find derivative using the definition x2 6. Know all the rules for derivatives that we covered in the class such as addition, subtraction, product, quotient, power, and chain rules 7. Find derivatives of the following functions using rules/formulas for derivatives (a) Find the derivative of y = f(x) = ( 3x 2 + 2)(5x 9) (b) Find the derivative of y = f(x) = (x 3 + 1) x (c) Find the derivative of y = f(t) = t4 + 5t 3 2t + 1 t x + 1 (d) Find the derivative of y = x (e) Find the derivative of y = x + 1 x e x x 1 (f) Find the derivative of y = x x + 1 x (g) Find the derivative of y = (x 3 + 1) 2 (h) Find the derivative of y = (x + 1)(2x 1)(x 3) (i) Find the derivative of y = x2 x + 1 (j) Find the derivative of y = tan x (k) Find the derivative of y = e x sin x cos x (l) Find the derivative of y = ln (x) sin x (m) Find the derivative of y = 3 (3x 2 2x + 1) 8. Find the derivative of y = e x3 +9x 2 9. Find the derivative of y = ln (x 2 + 1) 10. Find the derivative of y = sin 2 x + cos 2 x 11. Find the derivative of y = 5 sin 2 x 3 sin 2 x + cos x 12. Find the derivative of y = sin2 x sin x cos x 13. Find second derivative of y = f(x) = (x 3 + 3x) 2 2

3 14. Implicitly find dy dx and dx dy for 4x2 + 3y 2 = An object is thrown straight up with initial velocity of 160 feet per second from the ground. Find the follwing for this object: (a) Find x(t), the position of the object as a function of time (b) Find the velocity of the object as a function of time (c) Find the acceleration of the object as a function of time (d) Find the velocity and the acceleration of the object at time t = 2 (e) Find the highest point the object reaches (f) How long after the object is thrown, does it return to the ground? 16. You invest $20,000 in a new business venture. Your monthly revenue is $1,500 and monthly cost of doing business is $1,000; how long would it take to break even in this business at this rate? 17. Demand and Supply: For a consumer, quantity demanded decreases as price increases. But for a supplier, it is just the opposite; the quantity supplied increases as price increases: (a) Find the equilibrium price, the price at which the demand meets the supply, in a free market system given that demand curve is p = x and the supply curve is p = (b) Sketch the general shape of the demand and supply curve 18. Straight Line Depreciation: In business, it is common to depreciate an item for tax purposes. Typically straight line depreciation is easier and more common. You can depreciate an item yearly over its usable life time. Lets say you bought a piece of real estate, an office building, as part of your business; typically you would depreciate it over 27.5 years. If you bought the office building for $500,000 (a) Find the straight line depreciation model for this office building (b) For tax purposes, how would you value this office building after 10 years? 19. Revenue Model: For local small business, its yearly revenue is approximately modeled by R(t) = 2t 2 10t+131 where t is in years and revenue is in thousands of dollars. How is the revenue changing for this business at the end of the third year? 20. Marginal Cost: In Bangladesh, a young entrepreneur started a garments factory with initial investment of $2 million. Once the factory is established, the cost of making a T-shirt is $1.5. Create a cost model for this new venture and find the marginal cost. 3

4 21. Marginal Profit: Profit for a cell phone accessories business owner is given by P (x) = 0.05x 2 + 5x. Find the marginal profit for this business when quantity sold is Demand and Revenue: There are some businesses for which demand is cyclical, meaning that demand follows a pattern based on the time of the year or the the phase of the cycle. For example, demand for luxary items coincides with economic cycles; during a bull market, the demand for these items go up, and during a bear market, the demand goes down. Let us assume that demand for such a cyclic business is given by D(t) = 10 sin t + 20, where t is in years and D(t) is the quantity demanded as a function of time. Surprisingly, the prices for such big purchase items do not change much based on the demand; it is rather the economic cycle that determines the supply and demand. For example, a big purchase item such as car, it price does not change by that much, but its demand depends on economic cycle. Assume that a car that has price tag of $40,000. What is marginal revenue for the sellers of this car when t is 5? 4

5 Business Calculus: Functions Function: A function is a mapping between two sets with the condition that each item in the first set is mapped to only one item in the second. The first set is called the domain and the second set is called the range. A function is typically denoted by f(x) which means that the function f is dependent on the variable x. We write a function in the form y = f(x) where y is the dependent variable that depends on x and x is the independent variable. Problem 1: Find the domain and range of the real valued function y = f(x) = 2x + 3 Problem 2: Find the domain and range of the real valued function 6 y = f(x) = 1 x y = f(x) = 1 x Problem 3: Find the domain and range of the real valued function y = f(x) = 2 x 2 1 Problem 4: Find the domain and range of the real valued function y = f(x) = x 1 5

6 y = f(x) = x Problem 5 Write this piecewise function in algebraic form: f(x) = 3 4 f(x) = x 2 f(x) 6 = Problem 6: Sketch this function and find the domain and the range of the function: 1 : x [, 1] f(x) = 2 : x (1, 2) 0 : x [2, ) Problem 7: This step function is sometimes also called the Heaviside function where c is some constant. Sketch this function. f(x) = { 0 : x < c 1 : x c 6

7 Translation of Functions: Horizontal Translation: A function y = f(x) translated/shifted/moved by a (constant) units horizontally if the input x to the function is repaced by (x + a) i.e., y = f(x) y = f(x + a) It may be counter to your intuition that if a is positive, then the shift is to the left by a units, and if a is negative then the shift is to the right by a units. This shift is caused by an inside change to the function because the change takes place inside the function as in the input of the function y = f(x) = (x 1) 2 2 y = f(x) = x Figure: The horizontal shift of a parabola Problem 8: Sketch the translation of the function: y = f(x) = x 2 y = f(x) = (x 3) 2 Problem 9: Sketch the translation of the function: y = f(x) = x 3 y = f(x) = (x + 2) y = f(x) = x

8 Vertical Shift: A function y = f(x) shifted/moved b units vertically if a constant b is added to the function, that is y = f(x) y = f(x) + b If b is positive, then the function moves up by b units, and if b is negative then the function moves down by b units. The vertical shift is caused by an outside change to the function, because the change takes place outside of the function as caused by adding the constant b y = f(x) = x y = f(x) = x Figure: The vetical shift of a parabola Problem 10: Sketch the translation of the function: y = f(x) = x 2 y = f(x) = x 2 3 Problem 11: Sketch the translation of the function: y = f(x) = x 3 y = f(x) = x y = f(x) = x

9 Composition of Functions Function Composition: The composition of two functions f(x) and g(x) is denoted by f g and defined as f g(x) = f(g(x)) where x is in the domain of g such that g(x) in the domain of f. The key point here is to note that you give input x to the function g that gives you an output g(x) and that is the input to the function f as given by f(g(x)). To get a better understanding of function composition you would need to look at functions as input and output. A function can be viewed as a black box that gives unique output each unique input. Problem 12: Find the function composition f g given that f(x) = 2x + 3 g(x) = x 2 1 Problem 13: Find the function composition g f given that f(x) = 2x + 3 g(x) = x 2 1 Problem 14: Find the function composition f g given that f(x) = e x g(x) = 2x Problem 15: Find the function composition g f given that f(x) = e x g(x) = 2x Problem 16: Find the function self composition of the function f f given that f(x) = e x 9

10 Inverse Functions Inverse Functions: Two functions f and g are inverses of each other if f g(x) = g f(x) for all x values for which both f and g are defined or equivalently x is in the domains of both f and g. Problem 17: Show that the functions f and g are inverses of each other by showing that f g(x) = g f(x) given that f(x) = 2x + 3 g(x) = x 3 2 Problem 18: Show that the functions f and g are inverses of each other by showing that f g(x) = g f(x) given that f(x) = x g(x) = x 3 Problem 19: Show that the functions f and g are inverses of each other by showing that f g(x) = g f(x) given that f(x) = e x g(x) = ln(x) 1. Find limit lim x 1 (2x + 1) 2. Find limit lim x 0 (2x 2 + 1) 3. Find limit lim x 1 (5x 3 3x 2 2x + 1) 4. Find limit lim x 1 x 2 1 (x 1) 5. Find limit lim x 2 x 3 8 (x 2) 6. Find limit lim x 1 x + 1 x Find limit lim x 0 x x 8. Find limit lim x 0 x x Limits 10

11 9. Find limit lim x 1 2 x f(x) = { x : x (, 0) x 2 : x / (, 0) Rules of Limits 1. Scalar multiplication: lim x a [cf(x)] = al 2. Addition: lim x a [f(x) + g(x)] = lim x a [f(x)] + lim x a [g(x)] = L f + L g 3. Subtraction: lim x a [f(x) g(x)] = lim x a [f(x)] lim x a [g(x)] = L f L g 4. Product: lim x a [f(x) g(x)] = lim x a [f(x)] lim x a [g(x)] = L f L g 5. Quotient: lim x a f(x) g(x) = lim x a f(x) lim x a g(x) = L f L g given that L g 0 6. Power: lim x a [f(x)] n = L n 7. Radical: lim x a n f(x) = n L where f(x) under the radical should not cause complex numbers under radical Find limit in the following problems if it exists. If the limit does not exist, then justify/explain your answer: 1. lim x 1 5x 2 2. lim x 1 (3x 2 + 2x) 3. lim x 1 (4x 3 2x 2 + 7) 4. lim x 1 (x 2) (2x + 1) 5. lim x 1 (x 2) (2x + 1) 6. lim x 27 3 x 7. lim x 2 3 (x 2 + 2x) 8. lim x 0 (x + 3) 4 9. lim x x (x + 3) 10. lim x 0 x 2 x 12 x

12 11. lim t 0 t 2 10t + 9 2t 2 t lim t 0.5 t 2 6t + 9 2t Given that f(x) = x find the limit lim h 0 f(x + h) f(x) h 14. Given that f(x) = 5 find the limit lim h 0 f(x + h) f(x) h 15. Given that f(x) = 5x find the limit lim h 0 f(x + h) f(x) h 16. Given that f(x) = x 2 find the limit lim h 0 f(x + h) f(x) h 17. Given that f(x) = x 3 find the limit lim h 0 f(x + h) f(x) h 18. Given that f(x) = x 2 + x find the limit lim h 0 f(x + h) f(x) h 19. Given that f(x) = 1 x 2 find the limit lim h 0 f(x + h) f(x) h 20. If j(x) and g(x) are functions and f(x) = j(x)+g(x), then find lim h 0 f(x + h) f(x) h for f(x) 21. If j(x) and g(x) are functions and f(x) = j(x) g(x), then find lim h 0 f(x + h) f(x) h for f(x) 12

13 Derivatives Do the following deirvatives related problems: 1. Show that the derivative of a constant is zero using the definition lim h 0 f(x + h) f(x) h 2. Find the derivative of y = f(x) = 5x 2 using the definition of derivative lim h 0 f(x + h) f(x) h 3. Find the derivative of y = f(x)+g(x) using the definition of derivative lim h 0 f(x + h) f(x) h 4. Find the derivative of y = (2x + 1)(x 7) 5. Find the derivative of y = ( 3x 2 + 2)(5x 9) 6. Find the derivative of y = (x 3 + 1) x 7. Find the derivative of y = (x 3 + 1) x 8. Find the derivative of y = x + 1 x 1 9. Find the derivative of y = x + 1 3x Find the derivative of y = t4 + 5t 3 2t + 1 t Find the derivative of y = 3 64x + 1 x 12. Find the derivative of y = x + 1 x e x 13. Find the derivative of y = x 1 x x + 1 x 14. Find the derivative of y = (x 3 + 1) Find the derivative of y = (x + 1)(2x 1)(x 3) 16. Find the derivative of y = x2 x Find the derivative of y = sin x 13

14 18. Find the derivative of y = cos x 19. Find the derivative of y = tan x 20. Find the derivative of y = e x sin x cos x 21. Find the derivative of y = ln (x) sin x 22. Find the derivative of y = e 2x Find the derivative of y = e x3 +9x Find the derivative of y = ln (x 2 + 1) 25. Find the derivative of y = ln (x) e x 26. Find the derivative of y = sin 2 x + cos 2 x 27. Find the derivative of y = 5 sin 2 x 3 sin 2 x + cos x 28. Find the derivative of y = sin2 x sin x cos x 29. Find the derivative of y = (2x + 1) Find the derivative of y = (x 4 5x 3 + x 2 7x + 1) Find the derivative of y = 4 x 3 + 5x 2 3x Find the derivative of y = ln (x 2 + 1) 33. Find the derivative of y = e (x2 +1) 34. Find the derivative of y = 3 (3x 2 2x + 1) 35. Find the derivative of y = 5 (2x 2 + 1) Find the derivative of y = x2 + 1 x 37. Find the derivative of y = x2 1 x Find the derivative of y = (x + 1) 3 (2x 2 + 9) 39. Find the derivative of y = (x 2 + 1) 4 (x 5) 14

15 40. Find the derivative of y = 41. Find y if y = (x 2 + 2x 7) Find y if y = x 2 + 2x 7) 43. Find y if x 2 y + y = Find y if x 2 y 7e xy = 2 ( x 2 1 ) 3 x

16 Business Calculus - Quiz 1 Name: Problem 1: Find the derivative dy dx of y = xex 5x 3 16

17 Combining Functions Functions can be combined in many different ways. But to begin with, we may apply the usual arithmetic operations to functions with certain restirctions. We can combine them using our familiar operations such as addition, subtraction, multiplication, and division. 17

18 If f(x) and g(x) are two functions with F and G as their corresponding domains, then the following operations are well defined on these functions: 1. Sum of f and g: (f + g)(x) = f(x) + g(x) where both f and g are defined. This new function will have domain D = F G 2. Difference of f and g: (f g)(x) = f(x) g(x) where both f and g are defined. This new function will have domain D = F G 3. Product of f and g: (f g)(x) = f(x) g(x) where both f and g are defined. This new function will have domain D = F G 4. Quotient of f and g: ( f g )(x) = f(x) g(x) where both f and g are defined. The domain of this new function will have the additional condition that g(x) 0 for any x in its domain because division by zero is not allowed. So, the domain of the quotient function is D = {x F18 G g(x) 0}

19 Note that these function operations can be extended to arbitrary number of functions, that is, just not only on two functions, but also to more than two functions. However, the most important thing to remember when combining functions is to ensure that the combined functions are well defined. This will be ensured, for combining functions through addition, subtraction, and multiplication, if we consider the new domain to be the intersection of the domains of the original functions. For division/quotient, we need the additional constraint that the denominator function cannot evaluate to zero for any value in the new domain. If these rules are followed, then have a well defined set of operations on functions that we can use to construct new and interesting functions from the given ones as the follwing examples describe. 19

20 Example 1: For functions f(x) = x 2 3x + 2 and g(x) = x 2 + 2x + 1, find f + g, f g, fg, f, that is, add, subtract, g multiply, and divide the functions, and find the domain of each of the new functions. Example 2: For functions f(x) = 1 x 3 and g(x) = 1 x + 5, find f + g, f g, fg, f, that is, add, subtract, multiply, and g divide the functions, and find the domain of each of the new functions. Example 3: For functions f(x) = x and g(x) = x 3, that is, add, subtract, multiply, and divide the functions, and find the domain of each of the new functions. Example 4: For functions f(x) = x 3 and g(x) = x + 5, that is, add, subtract, multiply, and divide the functions, and find the domain of each of the new functions. Example 5: For functions f(x) = x and g(x) = 5 x 3, that is, add, subtract, multiply, and divide the functions, and find the domain of each of the new functions. Example 6: Add and subtract the functions f(x) = ln(xy) and g(x) = lny 1, and find the domain of the new functiond given that x and y are positive real numbers, that is, {x, y R x > 0} 20

21 Composition of Functions Definition: The composition of two functions f and g is defined as (f g)(x) = f(g(x)) where the domain of f g consists all x s in the domain of g such that g(x) is in the domain of f. However, to get a better understanding of function composition, it is a great idea to look at function definition, and consquently function composition, from the perspective of an engineer. We look at functions as a black box that takes an input and gives only one output. Function, an engineering perspective: A function can be viewed as a black box that takes an input and gives only one output for each input. Example 1: Find f(x + 5) given that f(x) = 3x + 2 Example 2: Find f(x + 2) given that f(x) = x Example 3: Find f(x 1) given that f(x) = x 2 x + 3 Example 4: Find f(x π) given that f(x) = e x + e x Example 5: Find f(x 2 ) given that f(x) = xlnx 21

22 Example 6: Find f(x 2) given that f(x) = x 3 Example 7: Find f(x 2 + 2x) given that f(x) = x + 1 Function Composition Examples Example 8: Given that f(x) = 3x + 2 and g(x) = x 3 find the following function compsitions: f g, g f, f f, g g Example 9: Given that f(x) = 2x and g(x) = x + 1 find the following function compsitions: f g, g f, f f, g g Example 10: Given that f(x) = 3 x and g(x) = x find the following function compsitions: f g, g f, f f, g g Example 11: Given that f(x) = 3 x and g(x) = e x find the following function compsitions: f g, g f, f f, g g 22

23 Function Decomposition: So far We have put together functions in a special way to perform function composition, but we can also perform the reverse of this process. This means that we could decompose a function into multiple functions in such a way so that we can compose those functions to get the original function. However, keep in mind that the function decomposition is not unique and may have multiple ways to do so. Example 12: Decompose the function f(x) = 3 2x Example 13: Decompose the function f(x) = 2x One-to-one Functions 23

24 Definition: A function is one-to-one if each unique item in the domain is mapped to a unique item in the range. An equivalent definition of one-to-one function is that two different items in the domain is mapped to two different items in the range. This definition is more useful in verifying that a function is one-to-one. A third way to define one-to-one function is that a function is one-to-one if and only if f(x 1 ) = f(x 2 ) implies that x 1 = x 2. There is an easier way to determine whether a function is one-to-one, it is called the Horizontal Line Test. A function is one-to-one if and only if no horizontal line intersects the graph of the function at multiple locations, that is, if the horizontal line intersects the graph, then it must intersect at only one point on the graph. Example 14: Determine whether the function f(x) = 2x + 3 is one-to-one. Example 15: Determine whether the function f(x) = x is one-to-one. Example 16: Determine whether the function f(x) = x 3 is 24

25 one-to-one. Example 17: Determine whether the function f(x) = x 2 is one-to-one. Example 18: Determine whether the function f(x) = e x is one-to-one. Example 19: We define a function as even function if f( x) = f(x) for all x in its domain. Do you think that an even function is one-to-one? 25

26 Inverse Functions Definition: We define two functions f and g as inverses of each other if they satisfy f g(x) = x and g f(x) = x for the x in the corresponding domain of definition. A consequence of this definition is that if a function is not one-to-one, then it does not have an inverse. Hence a function must have to be one-to-one and onto for it to have an inverse. A function is called onto if every element in its range has some item mapped to it from the domain. Example 20: Find inverse of the function f(x) = 5x + 3 if it exists, and verify that the function you found is in fact the inverse of the given function. Example 21: Find inverse of the function f(x) = x + 3 if it exists, and verify that the function you found is in fact the inverse of the given function. Example 22: Find inverse of the function f(x) = x + 3 x + 1 if it exists, and verify that the function you found is in fact the inverse of the given function. Example 23: Find inverse of the function f(x) = x if it exists, and verify that the function you found is in fact the inverse of the given function. Example 24: Find inverse of the function f(x) = x + 3 if 26

27 it exists, and verify that the function you found is in fact the inverse of the given function. Example 25: Find inverse of the function f(x) = x if it exists, and verify that the function you found is in fact the inverse of the given function. 27