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Finding the Derivative by using Logarithmic Differentiation
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"Use logarithmic differentiation to find the derivative f’(x). f(x)=x�e^-x(3x 5)� "
Calculating the Principal Amount from the Given Information
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How much money should be invested today at an annual interest rate of 7% compounded continuously so that 20 years from now it will be worth $20,000?
Finding the Coefficient of an Algebraic Expansion
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Find the coefficient of a6b2 in (a2 – 2b)5.
Proving a Mathematical Relation using Mathematical Induction
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Prove using mathematical induction: n3 2n is divisible by 3 for all positive integers n.
Maximum/Minimum and Concavity of the given Function
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Graph the function f(x) = x ln(x 3) 10xe-3x^2. Calculate the first and second derivatives. On the graph, clearly show where concavity changes (if it does change). Also, use the second derivative test to determine the maximums and minimums.
Derivatives from the Rules of Differentiation
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"Use rules of differentiation to write f’(x). Do not simplify. a. f(x) = x4 2x2 b. f(x) = x2‐9/x3 3 c. f(x) = x2 sqrt x2 4 d. f(x) = (x2/x 1)3 "
Second Derivative for a Function
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"Find f’’(x) Do not simplify a. f (x) = x4‐5x2 7 b. f (x)= sqrt (x2 1) "
Finding the Slope of Tangent Line
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"Draw the graph of f(x) = 2 sqrt(x) a. What is the domain b. On what interval is it continuous c. Use the definition of the derivative to find f’(x) d. What is the domain f’(x) e. What is the slope of the tangent line to y = f(x) at x = 1 f. What is the slope at x = 4 g. Without calculating, decide whether the slope of the tangent line at x=9 is more or less than the slope at x = 4 h. Does lim x → 0 f(x) exist? i. Does lim x → 0 f’(x) j. Describe the behavior of f’(x) as x → o "
Rate of Change for the Given Function
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"1. An object moves according to the equation s(t) = 4t ‐ 32t. a. Find the average rate of change of s(t) over the interval t = a to t = a h. b. Find the average rate of change of s(t) over the interval t = 2 to t = 2 h. c. The instantaneous rate of change at t = a is lim h → 0 (average rate of change over t = a to t = a h) find the instantaneous rate of change of s(t) at t = 2. "
Finding the Number of Possibilities by using the Permutation and Combinations
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How many license plate numbers are there consisting of three letters followed by three digits?
Calculation of Current Stock Price
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"Based on the corporate valuation model, Morgan Inc.’s value of operations is $300 million. The balance sheet shows $90 million of notes payable, $30 million of long-term debt, $40 million of preferred stock, and $100 million of common equity. The company has 10 million shares of stock outstanding. What is the best estimate of the stock’s price per share? a. $12.00 b. $12.64 c. $13.30 d. $14.00 e. $14.70 "
Calculation of Current Stock Price
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"Schnusenberg Corporation just paid a dividend of D0 = $0.75 per share and that dividend is expected to grow at a constant rate of 6.50% per year in the future. The company's beta is 1.25, the required return on the market is 10.50%, and the risk-free rate is 4.50%. What is the company's current stock price? a. $14.52 b. $14.89 c. $15.26 d. $15.64 e. $16.03 "
Corporate Valuation using Constant Growth Model
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"Which of the following statements is CORRECT? a. To implement the corporate valuation model, we discount projected free cash flows at the weighted average cost of capital. b. To implement the corporate valuation model, we discount net operating profit after taxes (NOPAT) at the weighted average cost of capital. c. To implement the corporate valuation model, we discount projected net income at the weighted average cost of capital. d. To implement the corporate valuation model, we discount projected free cash flows at the cost of equity capital. The corporate valuation model requires the assumption of a constant growth rate in all years. "
Stock valuation and Comparison of Returns
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"Stocks A and B have the following data. The market risk premium is 6.0% and the risk-free rate is 6.4%. Assuming the stock market is efficient and the stocks are in equilibrium, which of the following statements is CORRECT? A B Beta 1.10 0.90 Constant growth rate 7.00% 7.00% a. Stock A must have a higher stock price than Stock B. b. Stock A must have a higher dividend yield than Stock B. c. Stock B’s dividend yield equals its expected dividend growth rate. d. Stock B must have the higher required return. e. Stock B could have the higher expected return. "
Stock valuation and Capital Gains yield
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"Stocks X and Y have the following data. Assuming the stock market is efficient and the stocks are in equilibrium, which of the following statements is CORRECT? X Y Price $30 $30 Expected growth (constant) 6% 4% Required return 12% 10% a. Stock X has a higher dividend yield than Stock Y. b. Stock Y has a higher dividend yield than Stock X. c. One year from now, Stock X’s price is expected to be higher than Stock Y’s price. d. Stock X has the higher expected year-end dividend. e. Stock Y has a higher capital gains yield. "
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