University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Exam 1 26 April 2013 Name: Problem 1 (25 points) Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Answer the following questions: a. Find cov(i; ^ 1). b. If 0 = 4 what is the least squares estimate of 1? c. What is the variance of the estimate of part (b)? d. Is the estimate of part (b) unbiased? Problem 2 (25 points) Answer the following questions: a. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ^ 0 is BLUE (it has the smallest variance among all the linear unbiased estimators of 0). b. Consider the model of part (a). Find cov(ei; ^ Yi). c. Consider the simple regression model through the origin yi = 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that Pn i=1 xiei where ei = Yi ? ^ Yi. d. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2xi, and cov(i; j) = 0. Also, x is nonrandom. Is the assumption of constant variance satis ed in the following model? Please explain. Yi p xi = 0 p xi + 1 xi p xi + i p xi : Problem 3 (25 points): Answer the following questions: a. Consider the model yi = 0 + 1xi +i. Assume that E(i) = 0, var(i) = 2, and cov(i; j) = 0. Suppose we rescale the x values as x = x ? , and we want to t the model yi =  0 +  1xi + i. Find the least squares estimates of  0 and  1 . b. Refer to the model yi =  0 +  1xi +i of part (a). Find the SSE of this model and compare it to the SSE of the model yi = 0 + 1xi + i. What is your conclusion? c. Consider the simple regression model yi = 0 + 1xi + i, with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ESY Y = (n ? 1)2 + 2 1SXX, where SY Y = Pn i=1(yi ? y)2 and SXX = Pn i=1(xi ? x)2. d. Refer to the model of part (c). Find cov(i; ei). Problem 4 (25 points) Suppose that a simple linear regression of miles per gallon (Y ) on car weight (x) has been performed on 32 observations. The least squares estimates are ^ 0 = 68:17 and ^ 1 = ?1:112, with se = 4:281. Other useful information: x = 30:91 and P32 i=1(xi ? x)2 = 2054:8. Answer the following questions: a. Construct a 95% con dence interval for 1. b. Construct a 95% con dence interval for 2. c. What is the value of R2? d. Construct a con dence interval for 3 0 ? 2 1 ? 50.

## University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Exam 1 26 April 2013 Name: Problem 1 (25 points) Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Answer the following questions: a. Find cov(i; ^ 1). b. If 0 = 4 what is the least squares estimate of 1? c. What is the variance of the estimate of part (b)? d. Is the estimate of part (b) unbiased? Problem 2 (25 points) Answer the following questions: a. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ^ 0 is BLUE (it has the smallest variance among all the linear unbiased estimators of 0). b. Consider the model of part (a). Find cov(ei; ^ Yi). c. Consider the simple regression model through the origin yi = 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that Pn i=1 xiei where ei = Yi ? ^ Yi. d. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2xi, and cov(i; j) = 0. Also, x is nonrandom. Is the assumption of constant variance satis ed in the following model? Please explain. Yi p xi = 0 p xi + 1 xi p xi + i p xi : Problem 3 (25 points): Answer the following questions: a. Consider the model yi = 0 + 1xi +i. Assume that E(i) = 0, var(i) = 2, and cov(i; j) = 0. Suppose we rescale the x values as x = x ? , and we want to t the model yi =  0 +  1xi + i. Find the least squares estimates of  0 and  1 . b. Refer to the model yi =  0 +  1xi +i of part (a). Find the SSE of this model and compare it to the SSE of the model yi = 0 + 1xi + i. What is your conclusion? c. Consider the simple regression model yi = 0 + 1xi + i, with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ESY Y = (n ? 1)2 + 2 1SXX, where SY Y = Pn i=1(yi ? y)2 and SXX = Pn i=1(xi ? x)2. d. Refer to the model of part (c). Find cov(i; ei). Problem 4 (25 points) Suppose that a simple linear regression of miles per gallon (Y ) on car weight (x) has been performed on 32 observations. The least squares estimates are ^ 0 = 68:17 and ^ 1 = ?1:112, with se = 4:281. Other useful information: x = 30:91 and P32 i=1(xi ? x)2 = 2054:8. Answer the following questions: a. Construct a 95% con dence interval for 1. b. Construct a 95% con dence interval for 2. c. What is the value of R2? d. Construct a con dence interval for 3 0 ? 2 1 ? 50.

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1. Isolate GT1 in the equation ΔGT2 T2 − ΔGT1 T1 = ΔH ( 1 T2 − 1 T1 ). 2. True or False? ln (a + b) = ln a + ln b … prove your answer numerically. 3. What is the base-e logarithm of 250, ln 250? Prove that your result works numerically. 4. Solve for x in the following equation: e–ax = 1/T. 5. Simplify the right side of the function y = eAeC/eT and then use the ln to solve for T. 6. At what values of ϕ are the functions sin(ϕ) or cos(ϕ) equal to 0? At what values are they each equal to 1? 7. Linearize the following equation to find m and K from the slope and intercept: v = m[X]/(K + [X]). 8. Find the 1st and 2nd derivatives of y(x) = 3×4 – 2×2 + 15. 9. Identify the locations of minima and maxima for the function given in the problem above. 10. Find the derivative of the function y(x) = 3 ln (2×2). Ψ(x), [Ψ(x)]2, Ψ’(x), and Ψ’’(x). 12. Integrate 3/x from to 1 to 3. 13. What is the integral of 3×2 – 2x + 4 between -1 and 1? 14. Integrate the following function from 2 to 4: y = 3 cos x + x/2. (Don’t forget to check your calculator settings! Is it set for degrees or radians?) 15. Isolate like terms and integrate both sides of your resulting differential equation: dy/dx = – 4 x-1 y-3.

## 1. Isolate GT1 in the equation ΔGT2 T2 − ΔGT1 T1 = ΔH ( 1 T2 − 1 T1 ). 2. True or False? ln (a + b) = ln a + ln b … prove your answer numerically. 3. What is the base-e logarithm of 250, ln 250? Prove that your result works numerically. 4. Solve for x in the following equation: e–ax = 1/T. 5. Simplify the right side of the function y = eAeC/eT and then use the ln to solve for T. 6. At what values of ϕ are the functions sin(ϕ) or cos(ϕ) equal to 0? At what values are they each equal to 1? 7. Linearize the following equation to find m and K from the slope and intercept: v = m[X]/(K + [X]). 8. Find the 1st and 2nd derivatives of y(x) = 3×4 – 2×2 + 15. 9. Identify the locations of minima and maxima for the function given in the problem above. 10. Find the derivative of the function y(x) = 3 ln (2×2). Ψ(x), [Ψ(x)]2, Ψ’(x), and Ψ’’(x). 12. Integrate 3/x from to 1 to 3. 13. What is the integral of 3×2 – 2x + 4 between -1 and 1? 14. Integrate the following function from 2 to 4: y = 3 cos x + x/2. (Don’t forget to check your calculator settings! Is it set for degrees or radians?) 15. Isolate like terms and integrate both sides of your resulting differential equation: dy/dx = – 4 x-1 y-3.

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11. Define mechanical work and provide both an equation and the proper units for this quantity. 12. If the gauge pressure of a tire is 5 atm, what is the total pressure inside the tire? (hint: not 5 atm!) 13. Express the number of seconds in 1 year in scientific notation using units of kilo- and Megaseconds. 14. Express the average diameter of a human hair (Google!) in feet and meters (again, Sci. Notation!). 15. Convert your answers from #14 above into deci-, centi-, milli-, and micrometers. 16. For what functions, y(x), is the relationship dy/dx = Δy/Δx always true? 17. Seperate log(xn/y) into simple log form with no exponents. 18. Differentiate the functions y(x) = 4×3 + 3×2 + 2x + 1, f(x) = ln (x3), and P(r) = 14 e2r + 3. 19. Differentiate the functions y(x) = 3 sin 2x, f(x) = –2 cos x2, and Pr(x) = A sin2 kx. 20. What is the inverse of frequency? What are SI units of frequency and inverse frequency?

## 11. Define mechanical work and provide both an equation and the proper units for this quantity. 12. If the gauge pressure of a tire is 5 atm, what is the total pressure inside the tire? (hint: not 5 atm!) 13. Express the number of seconds in 1 year in scientific notation using units of kilo- and Megaseconds. 14. Express the average diameter of a human hair (Google!) in feet and meters (again, Sci. Notation!). 15. Convert your answers from #14 above into deci-, centi-, milli-, and micrometers. 16. For what functions, y(x), is the relationship dy/dx = Δy/Δx always true? 17. Seperate log(xn/y) into simple log form with no exponents. 18. Differentiate the functions y(x) = 4×3 + 3×2 + 2x + 1, f(x) = ln (x3), and P(r) = 14 e2r + 3. 19. Differentiate the functions y(x) = 3 sin 2x, f(x) = –2 cos x2, and Pr(x) = A sin2 kx. 20. What is the inverse of frequency? What are SI units of frequency and inverse frequency?

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These two are also similar. However, their factored forms are different. Where is the difference in the trinomial and where is the difference in their factored forms? How do you remember which is which? x^2 – 8x + 16 x^2 + 8x + 16

## These two are also similar. However, their factored forms are different. Where is the difference in the trinomial and where is the difference in their factored forms? How do you remember which is which? x^2 – 8x + 16 x^2 + 8x + 16

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1. The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.2 seconds and a standard deviation of 0.1 seconds. 1‐1. (2 points) What is the probability that a reaction requires more than 0.5 seconds? 1‐2. (2 points) What is the probability that a reaction requires between 0.4 and 0.5 seconds? 1‐3. (2 points) What is the reaction time that is exceeded 95% of the time? 2. Spherical Uniform Distribution (Google! You do not have to explain why): 2‐1. (2 points) How can we pick a set of random points uniformly distributed on the unit circle x12 + x 2=1? 2‐2. (2 points) How can we pick a set of random points uniformly distributed on the 4‐dimensional unit 2 2 2 2 2 sphere x1 + x2 + x3 + x4 + x5 =1? 3. The random variable X has a binomial distribution with n = 19 and p = 0.4. Determine the following probabilities. (You may use computer. But, you have to show the formula.) 3‐1. (2 points) P(X ≤ 12) 3‐2. (2 points) P(X ≥ 18) 3‐3. (2 points) P(13 ≤ X < 15) 4. (2 points) Show the mean and the variance of the triangular distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. (You must show why.) 5. (2 points) An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 6. Consider the following system made up of functional components in parallel and series. C2 0.80 C1 0.90 C4 0.95 C3 0.85 6‐1. (2 points) What is the probability that the system operates? 6‐2. (2 points) What is the probability that the system fails due to the components in series? Assume parallel components do not fail. 6‐3. (2 points) What is the probability that the system fails due to the components in parallel? Assume series components do not fail. 6‐4. (2 points) Compute and compare the probabilities that the system fails when the probability that component C1 functions is improved to a value of 0.95 and when the probability that component C2 functions is improved to a value of 0.85. Which improvement increases the system reliability more? 7. (2 points) Suppose that the joint distribution of X and Y has probability density function f(x, y) = 0.25xy for 0 < x < 2 and 0 < y < 2. Compute V(2X + 3Y). (Show all your work.)

## 1. The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.2 seconds and a standard deviation of 0.1 seconds. 1‐1. (2 points) What is the probability that a reaction requires more than 0.5 seconds? 1‐2. (2 points) What is the probability that a reaction requires between 0.4 and 0.5 seconds? 1‐3. (2 points) What is the reaction time that is exceeded 95% of the time? 2. Spherical Uniform Distribution (Google! You do not have to explain why): 2‐1. (2 points) How can we pick a set of random points uniformly distributed on the unit circle x12 + x 2=1? 2‐2. (2 points) How can we pick a set of random points uniformly distributed on the 4‐dimensional unit 2 2 2 2 2 sphere x1 + x2 + x3 + x4 + x5 =1? 3. The random variable X has a binomial distribution with n = 19 and p = 0.4. Determine the following probabilities. (You may use computer. But, you have to show the formula.) 3‐1. (2 points) P(X ≤ 12) 3‐2. (2 points) P(X ≥ 18) 3‐3. (2 points) P(13 ≤ X < 15) 4. (2 points) Show the mean and the variance of the triangular distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. (You must show why.) 5. (2 points) An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 6. Consider the following system made up of functional components in parallel and series. C2 0.80 C1 0.90 C4 0.95 C3 0.85 6‐1. (2 points) What is the probability that the system operates? 6‐2. (2 points) What is the probability that the system fails due to the components in series? Assume parallel components do not fail. 6‐3. (2 points) What is the probability that the system fails due to the components in parallel? Assume series components do not fail. 6‐4. (2 points) Compute and compare the probabilities that the system fails when the probability that component C1 functions is improved to a value of 0.95 and when the probability that component C2 functions is improved to a value of 0.85. Which improvement increases the system reliability more? 7. (2 points) Suppose that the joint distribution of X and Y has probability density function f(x, y) = 0.25xy for 0 < x < 2 and 0 < y < 2. Compute V(2X + 3Y). (Show all your work.)

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The following equation can be used to compute values of y as a function of x: y = b  e?ax  sin(b  x)  (0:012  x4 ? 0:15  x3 + 0:075  x2 + 2:5  x) where a and b are parameters. Write the equation for implementation with MATLAB, where a = 2, b = 5, and x is a vector holding values from 0 to =24 in increments of x = =40. Employ the minimum number of periods (i.e., dot notation) so that your formulation yields a vector for y. In addition, compute the vector z = y2 where each element holds the square of each element of y. Combine x, y, and z into a matrix w, where each column holds one of the variables, and display w using the short g format. In addition, generate a labeled plot of y and z versus x. Include a legend on the plot (use help to understand how to do this). For y, use a 1:5-point, dashdotted red line with 14-point, red-edged white-faced pentagram-shaped markers. For z, use a standard-sized (i.e., default) solid blue line with standard-sized, blue-edged, green-faced square markers.

## The following equation can be used to compute values of y as a function of x: y = b  e?ax  sin(b  x)  (0:012  x4 ? 0:15  x3 + 0:075  x2 + 2:5  x) where a and b are parameters. Write the equation for implementation with MATLAB, where a = 2, b = 5, and x is a vector holding values from 0 to =24 in increments of x = =40. Employ the minimum number of periods (i.e., dot notation) so that your formulation yields a vector for y. In addition, compute the vector z = y2 where each element holds the square of each element of y. Combine x, y, and z into a matrix w, where each column holds one of the variables, and display w using the short g format. In addition, generate a labeled plot of y and z versus x. Include a legend on the plot (use help to understand how to do this). For y, use a 1:5-point, dashdotted red line with 14-point, red-edged white-faced pentagram-shaped markers. For z, use a standard-sized (i.e., default) solid blue line with standard-sized, blue-edged, green-faced square markers.

Prove that cosx=1-tan^2 x/2 / 1+tan^2 x/2

## Prove that cosx=1-tan^2 x/2 / 1+tan^2 x/2

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WEEKLY ASSIGNMENT #5 (WOW THAT TOOK A WHILE) NAME: 1. Find the linear approximation of the function f(x; y; z) = p x2 + y2 + z2 at some point to approximate a value of the number p (3:02)2 + (1:97)2 + (5:99)2. 1 2. Consider your favorite function, the Cobb-Douglas production function. P(L;K) = 1:5L:65K:35 modeling the production of the state of Idaho. Over time we discover that capitol is gradually increasing at an approximate rate of 0:02 units per year. If we decide as a group that we are perfectly happy with our production level and would rather have additional vacation time then how much can we decrease labor by each year and keep the same level of production. In how long(rounded up to the nearest year) will we have an additional week of vacation? 2 3. Use the chain rule to find dz dt or dw=dt (a) z = x?y x+2y x = et; y = e?t. (b) w = sin x cos x x = p t; y = 1=t. 4. Use the chain rule to find @z=@t or @z=@s (a) z = (x ? y)5 x = s2t; y = st2 (b) z = er cos  r = st;  = p x2 + y2. 3 5. The temperature at a point (x; y; z) is given by the function T(x; y; z) = 200e?x2?3y2?9z2 where T is measure in C and x; y; z in meters. (a) Find the rate of change of temperature at the point (2;?1; 2) in the direction toward the point (3;?3; 3). (b) In which direction does the temperature increase fastest, and what is that fastest rate? 4 6. Suppose (1; 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f. (a) fxx(1; 1) = 4; fxy(1; 1) = 1; fyy(1; 1) = 2 (b) fxx(1; 1) = 4; fxy(1; 1) = 3; fyy(1; 1) = 2 (c) fxx(1; 1) = ?1; fxy(1; 1) = 6; fyy(1; 1) = 1 (d) fxx(1; 1) = ?1; fxy(1; 1) = 2; fyy(1; 1) = ?8 (e) fxx(1; 1) = 4; fxy(1; 1) = 6; fyy(1; 1) = 9 5 Bonus Show that f(x; y) = x2 + 4y2 ? 4xy + 2 has an infinite number of critical points, and for all of them D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 6

## WEEKLY ASSIGNMENT #5 (WOW THAT TOOK A WHILE) NAME: 1. Find the linear approximation of the function f(x; y; z) = p x2 + y2 + z2 at some point to approximate a value of the number p (3:02)2 + (1:97)2 + (5:99)2. 1 2. Consider your favorite function, the Cobb-Douglas production function. P(L;K) = 1:5L:65K:35 modeling the production of the state of Idaho. Over time we discover that capitol is gradually increasing at an approximate rate of 0:02 units per year. If we decide as a group that we are perfectly happy with our production level and would rather have additional vacation time then how much can we decrease labor by each year and keep the same level of production. In how long(rounded up to the nearest year) will we have an additional week of vacation? 2 3. Use the chain rule to find dz dt or dw=dt (a) z = x?y x+2y x = et; y = e?t. (b) w = sin x cos x x = p t; y = 1=t. 4. Use the chain rule to find @z=@t or @z=@s (a) z = (x ? y)5 x = s2t; y = st2 (b) z = er cos  r = st;  = p x2 + y2. 3 5. The temperature at a point (x; y; z) is given by the function T(x; y; z) = 200e?x2?3y2?9z2 where T is measure in C and x; y; z in meters. (a) Find the rate of change of temperature at the point (2;?1; 2) in the direction toward the point (3;?3; 3). (b) In which direction does the temperature increase fastest, and what is that fastest rate? 4 6. Suppose (1; 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f. (a) fxx(1; 1) = 4; fxy(1; 1) = 1; fyy(1; 1) = 2 (b) fxx(1; 1) = 4; fxy(1; 1) = 3; fyy(1; 1) = 2 (c) fxx(1; 1) = ?1; fxy(1; 1) = 6; fyy(1; 1) = 1 (d) fxx(1; 1) = ?1; fxy(1; 1) = 2; fyy(1; 1) = ?8 (e) fxx(1; 1) = 4; fxy(1; 1) = 6; fyy(1; 1) = 9 5 Bonus Show that f(x; y) = x2 + 4y2 ? 4xy + 2 has an infinite number of critical points, and for all of them D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 6

WEEKLY ASSIGNMENT #2 YOU 1. Verify for the Cobb-Douglas production function P(L;K) = 1:01L:75K:25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by to double production? 1 2. Let F(x; y) = 1+ p 4 ? y2. Evaluate F(3; 1). Find and sketch the domain of F. Find the range of F. 2 3. Draw a contour map of the function showing several level curves. (a) g(x; y) = x2 ? y2 (b) s(x; y) = y=(x2 + y2) 3 4. Find the limit if it exists or show that the limit does not exist. You do not have to use the epsilon delta method so it will either be “obviously” continuous or you will have to show that it is not by finding two paths which give different results. (a) lim (x;y)!(2;?1) x2y + xy2 x2 ? y2 (b) lim (x;y)!(0;0) x4 ? 4y2 x2 + 2y2 (c) lim (x;y)!(0;0) xy p x2 + y2 4 5. The temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. What are the meaning of the partial derivatives @T=@t; @T=@x; @T=@y? Moscow lies at 46:73N; 117W. Suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm, and the air to the north and east is cooler. Would you expect fx(117; 4673; 9); fy(117; 4673; 9); ft(117; 4673; 9) to be positive negative or positive? 5 6. Find the first partial derivatives of the following functions. (a) f(x; y) = x4 + 5xy3 (b) g(x; y) = t2e?t (c) h(s; t) = ln(s + t2) (d) i(x; y) = x y (e) R(p; q) = arctan pq2 6 7. Find @z=@x and @z=@y for the following, assuming that f and g are differentiable single variable functions Hint: Your answer should use f0 and/or g0. z = f(x)g(y) ; z = f(x=y) 7

## WEEKLY ASSIGNMENT #2 YOU 1. Verify for the Cobb-Douglas production function P(L;K) = 1:01L:75K:25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by to double production? 1 2. Let F(x; y) = 1+ p 4 ? y2. Evaluate F(3; 1). Find and sketch the domain of F. Find the range of F. 2 3. Draw a contour map of the function showing several level curves. (a) g(x; y) = x2 ? y2 (b) s(x; y) = y=(x2 + y2) 3 4. Find the limit if it exists or show that the limit does not exist. You do not have to use the epsilon delta method so it will either be “obviously” continuous or you will have to show that it is not by finding two paths which give different results. (a) lim (x;y)!(2;?1) x2y + xy2 x2 ? y2 (b) lim (x;y)!(0;0) x4 ? 4y2 x2 + 2y2 (c) lim (x;y)!(0;0) xy p x2 + y2 4 5. The temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. What are the meaning of the partial derivatives @T=@t; @T=@x; @T=@y? Moscow lies at 46:73N; 117W. Suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm, and the air to the north and east is cooler. Would you expect fx(117; 4673; 9); fy(117; 4673; 9); ft(117; 4673; 9) to be positive negative or positive? 5 6. Find the first partial derivatives of the following functions. (a) f(x; y) = x4 + 5xy3 (b) g(x; y) = t2e?t (c) h(s; t) = ln(s + t2) (d) i(x; y) = x y (e) R(p; q) = arctan pq2 6 7. Find @z=@x and @z=@y for the following, assuming that f and g are differentiable single variable functions Hint: Your answer should use f0 and/or g0. z = f(x)g(y) ; z = f(x=y) 7

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