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13 Nov 2019
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evta) If a function f is continuous on the closed interval (a,b) and F is the antiderivative of fon the interval [a,bl, then f(x) F(b)- F(a). Exercise 4 (12 points) Evaluate each definite integral. 3 2169-2 ã4u-2 ).ui du e) (2t-sin2t + cos2t)dt
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Hubert Koch
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11 Feb 2019
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5. The Second Fundamental Theorem of Calculus. Let f be a continuous function on an interval [a, oo). Define the function F on (a, 0o) by Fr)-f(t)dt. Then F is an anti-derivative off. That is, F"(z) = f(x). (a) Let G(z)= / e-2t dt for z 20. what is G'(x). [lint: Use the Second Fundamental Theorem of Calculus along with the chain rule.] (b) Find the exact value of /e -2tdt Hint: Define F(r) ãe-2tdt. Show that F(a) (1), then take the limit of F(a) as r approaches infinity
This Question: 1 pt 12 of 44 (0 complete) This Quiz: 44 pts possible Verify Property 2 of the definition of a probability density function over the given interval. ffx) 22x What is Property 2 of the definition of a probability density function? O A. The area under the graph of f over the interval [a,bl is b O B. The area under the graph of f over the interval [a.b] is 1 O C. The area under the graph of f over the interval [a,b is a. Since the given interval. [0.oo), has a right-endpoint of infinity, the area under the graph of fwill be calculated using an improper integral. Choose the correct formula for an improper integral over the interval [a,ool below t(x) dx= lim | f(x) dx= lim [F(x)]a= lim (F(b)-F(a)) b-o0 f(x) dx = lim | f(x) dx = lim [F(x)]-lim (F(a)-F(b)) b- 00 b--0o f(x) dk- lim x) dx- lim [F(x)m (F(b)-F(a)) b-+00
O D. 00 f(x) dx= lim f(x) dx= lim [F(棴= lim (F(a)-F(b)) b--oã Substitute a and b into the left side of the formula from the previous step and express the integral as a limit. area 2edx lim2edx Next, determine F(x). First, find the antiderivative of f. 2(Use C as the artbitrary constant.)
Next, determine F(x). First, find the antiderivative of f. as the anbliay cosat) Let C 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over [0,oo) using the far right side of the formula for the area. area = As b approaches oo, e approaches infinity where c is a constant. Use this fact to calculate the limits from the previous step area+1 Is Property 2 of the definition of a probability density function over the given interval now verified? Choose the correct answer below A Property 2 o the definition o a probability density unction over the given interval has been ver ed since the expression in the previous step simplifies to o B. Property 2 of the definition of a probability density function over the given interval has not been verified because the expression in the previous step does not simplify to the expected area value. ° C. â D Property 2 ofthe definition of a probability density function over the given interval has been verified since the expression in the previous step simplifies to 0 Property 2 o the definition o a probability density unction over the given interval has been ve ned since the expressio n e evious ste si plifies to 1.
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This Question: 1 pt 12 of 44 (0 complete) This Quiz: 44 pts possible Verify Property 2 of the definition of a probability density function over the given interval. ffx) 22x What is Property 2 of the definition of a probability density function? O A. The area under the graph of f over the interval [a,bl is b O B. The area under the graph of f over the interval [a.b] is 1 O C. The area under the graph of f over the interval [a,b is a. Since the given interval. [0.oo), has a right-endpoint of infinity, the area under the graph of fwill be calculated using an improper integral. Choose the correct formula for an improper integral over the interval [a,ool below t(x) dx= lim | f(x) dx= lim [F(x)]a= lim (F(b)-F(a)) b-o0 f(x) dx = lim | f(x) dx = lim [F(x)]-lim (F(a)-F(b)) b- 00 b--0o f(x) dk- lim x) dx- lim [F(x)m (F(b)-F(a)) b-+00
O D. 00 f(x) dx= lim f(x) dx= lim [F(棴= lim (F(a)-F(b)) b--oã Substitute a and b into the left side of the formula from the previous step and express the integral as a limit. area 2edx lim2edx Next, determine F(x). First, find the antiderivative of f. 2(Use C as the artbitrary constant.)
Next, determine F(x). First, find the antiderivative of f. as the anbliay cosat) Let C 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over [0,oo) using the far right side of the formula for the area. area = As b approaches oo, e approaches infinity where c is a constant. Use this fact to calculate the limits from the previous step area+1 Is Property 2 of the definition of a probability density function over the given interval now verified? Choose the correct answer below A Property 2 o the definition o a probability density unction over the given interval has been ver ed since the expression in the previous step simplifies to o B. Property 2 of the definition of a probability density function over the given interval has not been verified because the expression in the previous step does not simplify to the expected area value. ° C. â D Property 2 ofthe definition of a probability density function over the given interval has been verified since the expression in the previous step simplifies to 0 Property 2 o the definition o a probability density unction over the given interval has been ve ned since the expressio n e evious ste si plifies to 1.
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