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Business Calculus Review

(1) If a sum of money is invested for a time period (in years) at an interest rate per period, the simple interest is given by Solving for we have,  

    (a)         
    
    (b)         
    
    (c)     
    
    (d)       
    
    (e)
        

(2) Given , the value of   is    

    (a)  6        
    
    (b)       
    
    (c)       
    
    (d)            
    
    (e) not enough information
    

(3) What is the present value of an investment at 6% annual simple interest if it is worth 832 dollars in 8 months?

    (a)         
    
    (b)          
    
    (c)     
    
    (d)     
    
    (e) none of the above
    

(4) Solve for with

    (a)       
    
    (b)       
    
    (c)        
    
    (d)       
    
    (e) no solution
    

(5) Most people who are retiring purchase an annuity to supplement their income. This annuity pays them a fixed sum of money at regular intervals. The single sum of money required to purchase an annuity that will provide these payments is called the

    (a)  principal   
    
    (b)  down payment    
    
    (c) annual percentage yield   
    
    (d)  future value  
    
    (e) present value
    

(6) The function defined by is an exponential growth function.    

    (a) True, because         
    
    (b) False, because
    
    (c) False
    
    (d) True, because
    
    (e) none of the above
    

(7) Suppose a couple is saving for a future purchase by making periodic payments to produce a future sum on a specified date. This type of investment is called a    

    (a)  simple interest   
    
    (b)  decay function    
    
    (c)  sinking fund   
    
    (d)  loan  
    
    (e) amortization
    

(8) Suppose 100,000 dollars is invested for 10 years at 8% compounded quarterly. For the equation, , the present value is given by

     (a)     
     
     (b)         
     
     (c)      
     
     (d)         
     
     (e) none of the above
    

(9) If  10,000 dollars is invested at 1% per month, the future value at any time (in months) is given by   How long before the investment doubles?

(10) Develop an amortization schedule for a loan of 30,000 dollars to be repaid in 4 annual payments of equal size. The interest rate is 7%.

(11) Is it more economical to buy an automobile for 29,000 dollars in cash or to pay 8000 dollars down and 3000 dollars at the end of each quarter for 2 years, if money if worth 8% compounded quarterly?

    Solution. The automobile can be bought now for 29,000 dollars or can be bought for 8000 dollars plus the present value of the investment. The present value is given by the formula where and so we have Thus the automobile can be bought for 29,000 dollars or for dollars. Thus, it is cheaper to pay cash.

(12) Given determine

    (a)         
    
    (b) does not exist        
    
    (c)         
    
    (d)          
    
    (e) none of the above

Solution.   Since



and



We see The correct choice for is (c).

(13) Determine whether the function is continuous at

    (a) yes, because         
    
    (b) no, because does not exist
    
    (c) no, because does not exist.
    
    (d) yes, because
    
    (e) none of the above

Solution.   Since but so is not continuous because does not exist. The correct choice is (c).

(14) Find the average rate of change of over the interval

    (a) 7        
    
    (b) 6.9    
    
    (c)
    
    (d) 6.8
    
    (e) none of the above

Solution. The average rate of change of is

  

The correct choice is (b).

(15) Given the revenue function (in dollars) where denotes the number of units sold. What is the marginal revenue if 50 units are sold?

    (a) 290
    
    (b) 12500    
    
    (c) 198    
    
    (d) 15198         
    
    (e) none of the above

Solution. Since we see the marginal revenue for units sold is   The correct choice is (e).

(16) Find given

    (a)
    
    (b)     
    
    (c)         
    
    (d)
    
    (e) none of the above

Solution. Using the power rule, The correct choice is (b).

(17) Find given

    (a)         
    
    (b)         
    
    (c)          
    
    (d)              
    
    (e) none of the above

Solution. Using the power rule    The correct choice is (d).

(18) Find an equation of the tangent line to the graph of the function at

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above

Solution. First and when When we find Therefore, an equation of the tangent line is which simplifies to The correct choice is (d).

(19) Find given

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above

Solution. Using the quotient rule   The correct choice is (a).

(20) Find given

    (a)          
    
    (b)         
    
    (c)     
    
    (d)              
    
    (e) none of the above

Solution. Using the chain rule,



The correct choice is (d).

(21) Given the derivative of a function the second derivative test concludes that

    (a) and so is a relative minium point.
    
    (b) and so is a relative minium point.
    
    (c) and so is a relative maximum point.        
    
    (d)   and so is a relative maximum point.              
    
    (e) none of the above

Solution. Since and so which means is a relative minium point. The correct choice is (b).

(22) Find if

    (a)         
    
    (b)         
    
    (c)     
    
    (d)     
    
    (e) none of the above

Solution. We find   and so The correct choice is (d).

(23) Determine the intervals where the graph of the function is increasing given the first derivative of as

    (a) and     
    
    (b) and         
    
    (c)         
    
    (d)       
    
    (e) none of the above

Solution.



  The correct choice is (a).

(24) Determine where the relative maximum occur for the function whose first derivative is given as

    (a) at
    
    (b) at         
    
    (c) at         
    
    (d) there are no relative maximum for the function
    
    (e) none of the above

Solution.  


The correct choice is (a).

(25) Assume that a function has domain all real numbers and whose first and second derivatives are and respectively, determine where the points of inflection occur for the graph of the function

    (a) no points of inflection
    
    (b) at and at
    
    (c) at

    (d) at and at   
    
    (e) none of the above

Solution.


Thus, the and is where the points of inflection occur. The correct choice is (d).

(26) Assume that a function has domain all real numbers and whose first and second derivatives are and respectively, determine the intervals where the graph of the function is concave up.

    (a)            
    
    (b)   and      
    
    (c) and          
    
    (d) and      
    
    (e) none of the above

Solution.  


Thus, and are the intervals when the graph of is concave up.  The correct choice is (b).

(27) Given the second derivative test  

    (a) concludes that is a critical point of
    
    (b) concludes that has a local maximum at          
    
    (c) concludes that has a local minimum at                  
    
    (d) is inconclusive for          
    
    (e) none of the above

Solution.  Since the critical numbers are and Also, since and so Thus the second derivative test is inconclusive for The correct choice is (d).

(28) Given then

    (a)     
    
    (b)     
    
    (c)         
    
    (d)               
    
    (e) none of the above

Solution.  By definition, The correct choice is (a).

(29) Determine given

    (a)         
    
    (b)         
    
    (c)         
    
    (d)              
    
    (e) none of the above

Solution. Using the quotient rule the derivative of is



The correct choice is (e).

(30) Determine given

    (a)         
    
    (b)          
    
    (c)          
    
    (d)               
    
    (e) none of the above

Solution. Using the chain rule, the derivative of is





The correct choice is (c).

(31) The average rate of change of over the interval is

    (a)          
    
    (b)         
    
    (c)         
    
    (d)              
    
    (e) none of the above

Solution. The average rate of change of over the interval is so, the average rate of change of over is



The correct choice is (d).

(32) The marginal revenue of at is

    (a)     

    (b)     

    (c)

    (d)
    
    (e) none of the above

Solution. The marginal revenue of at is and so at we find the marginal revenue to be The correct choice is (b).

(33) Find the equation of the tangent line to the graph of the function when for

    (a)         
    
    (b)         
    
    (c)          
    
    (d)              
    
    (e) none of the above

Solution. The slope of the tangent line is because and when



The point is found by Thus an equation of the tangent line is , that is The correct choice is (a).

(34) Find when

    (a)         
    
    (b)         
    
    (c)         
    
    (d)              
    
    (e) none of the above

Solution. Using the power rule and chain rule to find,





The correct choice is (b).  

(35) Suppose the domain of is all real numbers except and then the critical numbers of are

    (a) and
    
    (b) and only
    
    (c) only
    
    (d) there are no critical points because the domain of is not all real numbers
    
    (e) none of the above

Solution. Since and both and are defined we see both and are critical numbers. However, even though is where is undefined, but is also not defined for so is not a critical number for The correct choice is (b).

(36) Assume the domain of a function is all real numbers and Apply the First Derivative Test to determine where the function is decreasing.

    (a) on the interval
    
    (b) on the intervals and
    
    (c) on the interval
    
    (d) on the intervals and
    
    (e) none of the above

Solution. The critical numbers are and When then When then When then and when then Therefore, is decreasing on the intervals and A sketch of a function with this derivative,



The the correct choice is (b).

(37) Given determine the intervals where is concave up.

    (a) and     
    
    (b)
    
    (c)             
    
    (d) and                   
    
    (e) none of the above

Solution. First when and When when When and so is concave up on the intervals and The correct choice is (a).

(38) If the total profit, in thousands of dollars, for a product is given by ESTIMATE the profit from the sale of the 16th unit using marginal profit.

    (a)     
    
    (b) half a penny
    
    (c)     
    
    (d)
    
    (e) none of the above

Solution. Since and so The units are thousands of dollars and so the correct choice is (d).

(39) Evaluate the definite integral

    (a) 0
    
    (b) 1
    
    (c) 2
    
    (d) 4
    
    (e) none of the above

(40) Evaluate the definite integral

    (a)
    
    (b)     
    
    (c)
    
    (d)
    
    (e) none of the above

(41) Find the value of the sum

    (a) 51
    
    (b) 10
    
    (c) 55
    
    (d)
    
    (e) none of the above

(42) Approximate the area under the curve by evaluating the function at the left-hand endpoints of the subintervals from to using 4 subintervals.

    (a) 10
    
    (b) 11.25
    
    (c)
    
    (d) 12
    
    (e) none of the above

(43) Evaluate the integral

    (a) where is a constant
    
    (b) where is a constant    
    
    (c) where is a constant
    
    (d) where is a constant
    
    (e) none of the above

(44) Evaluate the integral

    (a) where is a constant
    
    (b) where is a constant    
    
    (c) where is a constant
    
    (d) where is a constant
    
    (e) none of the above

(45) Evaluate the integral

    (a) where is a constant
    
    (b) where is a constant
    
    (c) where is a constant
    
    (d) where is a constant
    
    (e) none of the above

(46) Evaluate the integral

    (a) where is a constant
    
    (b) where is a constant
    
    (c) where is a constant
    
    (d) where is a constant
    
    (e) none of the above

(47) Evaluate the integral

    (a) where is a constant
    
    (b) where is a constant
    
    (c) where is a constant
    
    (d) where is a constant
    
    (e) none of the above

(48) If where is a constant then

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above

(49) Find the elasticity of the demand function at

    (a)
    
    (b)
    
    (c) 21
    
    (d)
    
    (e) none of the above

(50) Suppose the demand function for a product is given by How would revenue be affected by a price increase when and

    (a) A price increase will decrease the total revenue.
    
    (b) A price increase will increase the total revenue.
    
    (c) A price increase will not affect the total revenue.
    
    (d) not enough information
    
    (e) none of the above

(51) Find the derivative of with respect to the variable

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above

(52) Find the derivative of with respect to the variable

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above

(53) Find the equation of the tangent line to the graph of at

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above

(54) If possible, solve the equation

    Solution. Using properties of logarithms,
    














Since is not defined when there is no solution to the equation

(55) Simplify

    Solution. Using properties of logarithms,
    








Thus,

(56) A debt of 250000 dollars is amortized with 50 equal payments of . If interest is 4.3%, compounded monthly, the unpaid balance of the debt after 30 payments have been made, approximate to the nearest dollar, is

(57) If 22,000 dollars is invested at 5.8%, compounded continuously, how long will it take the investment to grow to 500,000 dollars?

    Solution. By the meaning of compounded continuously



So we solve for







years

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