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Unformatted text preview: ucsc econ/ams 11b Review Questions 6 fall 2008 Implicit diﬀerentiation and Taylor polynomials
1. Use implicit diﬀerentiation to ﬁnd the indicated derivative at the given point. dy at the point (1, 2) on the graph of the equation x3 y +2xy 3 − 4x2 y 2 = 2. dx du at the point (2, e) on the graph of the equation u2 ln v + v 2 e−u = 5. b. Find dv a. Find 2. Find the equation of the tangent line to the graph of the equation x3 y + 2xy 3 − 4x2 y 2 = 4 at the point (2, 1) on the graph. Note: You can use most of your work from problem 1a. here, but not all of it. 3. The demand equation for a ﬁrm’s product is given by p2 q + 2pq 3/2 + 3q 2 = 42500, where p is the price per unit for the ﬁrm’s product and q is the quantity demanded of the ﬁrm’s product. a. Find the price elasticity of demand for the ﬁrm’s product when p = 5 and q = 100. b. Estimate the percentage change in demand for the ﬁrm’s product, if they raise the price from p = 5 to p = 5.25. c. Use your answer to 3a to determine whether the ﬁrm’s revenue will increase or decrease if the ﬁrm raises the price? Explain you answer. dr d. Compute and use the approximation formula to estimate the change in dp p=5 the ﬁrm’s revenue if they raise the price from p = 5 to p = 5.25. 4. Compute the degree 10 Taylor polynomial of the function f (x) = ex , centered at the point x0 = 0. (This is easier than you may think at ﬁrst.) √ 5. Compute the degree 2 Taylor polynomial of the function g (x) = 3 x, centered at √ the point x0 = 1000. Use this polynomial to ﬁnd an approximate value for 3 1001. 6. Compute the degree 4 Taylor polynomial of the function y = ln x, centered at x = 1. Use this polynomial to compute the approximate values of ln 0.8 and ln 1.25. Note: You only need to use the Taylor polynomial once. 7. Compute the degree 2 Taylor polynomial of the function f (x) = √ x0 = 100. Use the polynomial that you found to estimate 102. √ x centered at ...
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- Spring '08