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APPM 1350 Midterm: appm1350fall2014exam1_sol

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turquoisegnu128

31 Jan 2019

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Fall 2014: for this problem, suppose f (x) = x 6 and g(x) = |2x 1|. (a) (6 pts) write down the domain and range of (f g)(x). Show all work. (b) (6 pts) evaluate the limit: lim x 0. 5 g(x) 1 2x (c) (6 pts) suppose we let h(x) =(f (x), if x > 6 g(x), if x 6. , are there any values of x for which h(x) is not continuous? (d) (7 pts) use the limit de nition of the derivative to nd f (106). = lim x 0. 5+ x 0. 5 lim x 0. 5+ lim x 0. 5 x 0. 5 . = does not exist. (c)(6 pts) note that for x 6= 6, h(x) is continuous since both f (x) and g(x) are well de ned and continuous. At x = 6, we need lim x 6+ h(x) = h(6) for continuity. Now note that h(x) = lim x 6 lim x 6+ h(x) = lim x 6+

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Related Questions

4. (21 pts) (I) (6 pts) Determine the value of a constant b for which the function g is continuous at 0. Show your work. Justify your answer! 2 x +b if x<0, X-5 g(x) = if x 20, x#4. We have to show that lim g (x) = g (0). g(0) = 0 +18 = -1; lim X0* 0 + 16 ) = lim x + 16 **0* x2 - 160 - 16- Since lim g (x) has to exist (in order for g to be continuous at x = 0), it has to be true that x-0- x0+ X- 0 lim g(x) - lim g(x) = lim g(x). Therefore, -1, and , so b = 5. so, if b = 5, lim g (x) = g(0), and therefore, g is continuous at x = 0. (II) (4 pts) Evaluate the limit. Show your work! 2. b limx- (a) lim g(x) = lim *+- 2 x + b X - 5 : 1 * +16 10 g (x) = lim x++ - Elim x2 - 16 x++ (x - 10). OR 2x+5. 2; (a) lim g (x) = lim x--00 X- (b) lim x++ 2 + 16 g(x) = lim **** x2 - 16 X-+ (III) (4 pts) Determine whether g has any horizontal asymptotes. If so, write the equation (or equations) of all horizontal asymptotes. y = 2, by part (II) (a) y = 0, by part (II) (b) (IV) (7 pts) Determine whether g has any vertical asymptotes. If so, write the equation (or equations) of all vertical asymptotes. Show your work. Justify your answer! 5 + 16 There is no vertical asymptote at x = 5, because lim g(x) = x+5 52-16' -8 + b (x) = there is no vertical asymptote at x = -4, because lim X -4 - 4 - 5 But, lim g(x) = lim + 16 x+4+ x2 - 16 lim_ (x + 16) -20 x+4+ (x - 4) 60 (x + 4) +8 (or the numerator goes to 20, and the denominator is positive and goes to o, as x + 4*) = +00 So, the function g has only one vertical asymptote, x = 4.

Algebra

The function f(x) = 4x/9x-7 is one-to-one. Find its inverse and check your answer. f -1 (x) = (Simplify your answer.) Graph the function. Based on the graph, state the domain and the range and find any intercepts. The domain is . (Type your answer in interval notation.) The range is . (Type your answer in interval notation.) Find the intercepts, if there are any. Select the correct choice below and fill in any answer boxes within your choice. The intercept(s) is'are . (Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer.) Solve the equation. What is the solution? x = (Use a comma to separate answers.) If f(x) = 5x2 + 3 and g(x) = 9x + a, find a so that the graph of f g crosses the y-axis at 8. What is the value of a? a = (Use reduced fractions or integers. Use commas to separate values.) The function f(x) = 7/4+x is one-to-one. Find its inverse and check your answer. f -1 (x) = (Simplify your answer.) Suppose that H(x) = (1/4)x - 4096 What is H( - 5)? What is the point that corresponds to this vahie on the graph of H? If H(x) = - 4032, what is x? What is the point that corresponds to this vahie on the graph of H? Find the zero of H. H( - 5) = (Type an integer or a simplified fraction.) The point that corresponds to this vahie on the graph of H is . (Type an ordered pair, but do not use commas in any individual coordinates.) If H(x) = - 4032, then x = . (Type an integer or a simplified fraction.) The point that corresponds to this vahie on the graph of H is . (Type an ordered pair, but do not use commas in any individual coordinates.) The zero of H is x = (Type an integer or a simplified fraction.) If f(x) = x3 - 3x2 + 2x - 9 and g(x) = 4: find (f g)(x) and (g f)(x). What is (f g)(x)? What is (f g)(x)? (f g)(x) = What is (g f)(x)? (g f)(x) = The function f(x) = 8x/x+5 is one-to-one. Find its inverse and check your answer. f -1 (x) = (Simplify your answer.) Solve the equation. ex= e9x + 72 x = (Use a comma to separate answers as needed.)

Show transcribed image text The function f(x) = 4x/9x-7 is one-to-one. Find its inverse and check your answer. f -1 (x) = (Simplify your answer.) Graph the function. Based on the graph, state the domain and the range and find any intercepts. The domain is . (Type your answer in interval notation.) The range is . (Type your answer in interval notation.) Find the intercepts, if there are any. Select the correct choice below and fill in any answer boxes within your choice. The intercept(s) is'are . (Type an ordered pair. Use a comma to separate answers as needed. Type an exact answer.) Solve the equation. What is the solution? x = (Use a comma to separate answers.) If f(x) = 5x2 + 3 and g(x) = 9x + a, find a so that the graph of f g crosses the y-axis at 8. What is the value of a? a = (Use reduced fractions or integers. Use commas to separate values.) The function f(x) = 7/4+x is one-to-one. Find its inverse and check your answer. f -1 (x) = (Simplify your answer.) Suppose that H(x) = (1/4)x - 4096 What is H( - 5)? What is the point that corresponds to this vahie on the graph of H? If H(x) = - 4032, what is x? What is the point that corresponds to this vahie on the graph of H? Find the zero of H. H( - 5) = (Type an integer or a simplified fraction.) The point that corresponds to this vahie on the graph of H is . (Type an ordered pair, but do not use commas in any individual coordinates.) If H(x) = - 4032, then x = . (Type an integer or a simplified fraction.) The point that corresponds to this vahie on the graph of H is . (Type an ordered pair, but do not use commas in any individual coordinates.) The zero of H is x = (Type an integer or a simplified fraction.) If f(x) = x3 - 3x2 + 2x - 9 and g(x) = 4: find (f g)(x) and (g f)(x). What is (f g)(x)? What is (f g)(x)? (f g)(x) = What is (g f)(x)? (g f)(x) = The function f(x) = 8x/x+5 is one-to-one. Find its inverse and check your answer. f -1 (x) = (Simplify your answer.) Solve the equation. ex= e9x + 72 x = (Use a comma to separate answers as needed.)