Area functions and the Fundamental Theorem Consider the function f ( t ) = { t i f − 2 ≤ t < 0 t 2 2 i f 0 ≤ t ≤ 2 and its graph shown below . Let F ( x ) = ∫ − 1 x f ( t ) d t and ∫ − 2 x f ( t ) d t . 56. a. Evaluate G (−1) and G (1). b. Use the Fundamental Theorem to find an expression for G′ ( x ), for −2 ≤ x ≤ 0. c. Use the Fundamental Theorem to find an expression for G′ ( x ), for 0 ≤ x ≤ 2. d. Evaluate G′ (0) and G′ (1). Interpret these values. e. Find a constant C such that F ( x ) = G ( x ) + C .
Area functions and the Fundamental Theorem Consider the function f ( t ) = { t i f − 2 ≤ t < 0 t 2 2 i f 0 ≤ t ≤ 2 and its graph shown below . Let F ( x ) = ∫ − 1 x f ( t ) d t and ∫ − 2 x f ( t ) d t . 56. a. Evaluate G (−1) and G (1). b. Use the Fundamental Theorem to find an expression for G′ ( x ), for −2 ≤ x ≤ 0. c. Use the Fundamental Theorem to find an expression for G′ ( x ), for 0 ≤ x ≤ 2. d. Evaluate G′ (0) and G′ (1). Interpret these values. e. Find a constant C such that F ( x ) = G ( x ) + C .
Solution Summary: The author evaluates the value of G(-1) and -32.
Question
Is the function f(x) shown in the graph below continuous at x = -5?
f(z)
7
6
5
4
2
1
0
-10
-6 -5
-4
1
0
2
3
5
7
10
-1
-2
-3
-4
-5
Select the correct answer below:
The function f(x) is continuous.
The right limit exists. Therefore, the function is continuous.
The left limit exists. Therefore, the function is continuous.
The function f(x) is discontinuous.
We cannot tell if the function is continuous or discontinuous.
The graph of f(x) is given below. Select all of the true statements about the continuity of f(x) at x = -1.
654
-2-
-7-6-5-4-
2-1
1 2
5 6 7
02.
Select all that apply:
☐ f(x) is not continuous at x = -1 because f(-1) is not defined.
☐ f(x) is not continuous at x = −1 because lim f(x) does not exist.
x-1
☐ f(x) is not continuous at x = −1 because lim ƒ(x) ‡ ƒ(−1).
☐ f(x) is continuous at x = -1
J-←台
Let h(x, y, z)
=
—
In (x) — z
y7-4z
-
y4
+ 3x²z — e²xy ln(z) + 10y²z.
(a) Holding all other variables constant, take the partial derivative of h(x, y, z) with
respect to x, 2 h(x, y, z).
მ
(b) Holding all other variables constant, take the partial derivative of h(x, y, z) with
respect to y, 2 h(x, y, z).
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