Which of the following equations correctly demonstrates the application of the Fundamental Theorem of Calculus to evaluate $$\int_{1}^{4} x^3 dx$$?
Using the first part of the FTC (antiderivatives) to evaluate definite integrals.
34 questions
Which of the following equations correctly demonstrates the application of the Fundamental Theorem of Calculus to evaluate $$\int_{1}^{4} x^3 dx$$?
$$x^3\Big|_{1}^{4} = 4^3 - 1^3$$
$$\frac{x^4}{4}\Big|_{1}^{4} = \frac{4^4}{4} - \frac{1^4}{4}$$
$$\frac{x^4}{4} + C\Big|_{1}^{4}$$
$$\frac{x^3}{3}\Big|_{1}^{4}$$
$$If g(x) = x \ln(x) - x + C and f(x) = g'(x), which of the following is equal to \int_{1}^{e} f(t) dt?$$
e·ln(e) - e + 1
e·ln(e) - 1·ln(1)
e - 1
ln(e) - ln(1)
Which of the following is an example of correctly applying the First Fundamental Theorem of Calculus to evaluate the definite integral $$\int_{2}^{3}2*x dx$$?
Evaluate using the antiderivative $$F(x)=x^2$$ to obtain $$F(3)-F(2)=3^2-2^2=9-4=5$$.
Differentiate $$2*x$$ to get $$2$$ and then compute $$2*(3-2)=2$$.
Compute the sum of the function values at x=2 and x=3, giving $$2*3+2*2=6+4=10$$.
Approximate the area by using a rectangle with height $$2*2.5$$ and width 1 to get 5.
Let F be the function defined by F(x) = $\displaystyle\int_{1}$ˣ sin(t²) dt. What is an equation of the line tangent to the graph of F at x = 2?
y = sin(2²)(x - 2) + $\displaystyle\int_{1}^{2}$ sin(t²) dt
y = sin(4)(x - 2) + F(2)
y = 4cos(4)(x - 2) + F(2)
y = 2sin(4)(x - 2) + F(2)
If $$\frac{d}{dx}\left(\int_{0}^{x} (t^2+1) dt\right) = 10$$, what is the value of $$x$$?
$$3+\sqrt{0}$$
$$9$$
$$3$$
$$-3$$
If a function $$f$$ is continuous on the interval $$[a, b]$$, which of the following statements regarding the Fundamental Theorem of Calculus is true?
The Fundamental Theorem of Calculus requires summing all values of $$f(x)$$ without involving an antiderivative.
If $$F(x)$$ is an antiderivative of $$f(x)$$, then $$\int_{a}^{b} f(x) dx = F(a) - F(b)$$, reversing the proper order.
The Fundamental Theorem of Calculus states that $$\int_{a}^{b} f(x) dx$$ equals the average value of $$f(x)$$ over $$[a,b]$$.
If $$F(x)$$ is an antiderivative of $$f(x)$$, then $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$.
Let the function $$S$$ be defined by $$ S(x) = \int_{1}^{x} \left( \arctan(2t) + e^{-t} \right) dt $$. Which of the following is an expression for $$S'(x)$$?
$$ \frac{2}{1+4x^2} - e^{-x} $$
$$ \frac{1}{2}x\arctan(2x) - xe^{-x} $$
$$ \arctan(2x) + e^{-x} $$
$$ \frac{1}{1+4x^2} + e^{-x} $$
Let $$H$$ be the function defined by $$ H(x) = \int_{1}^{x} \left( \sqrt{t} + \frac{2}{t} \right) dt $$. Which of the following is $$H'(x)$$?
$$ \frac{1}{2\sqrt{x}} + \frac{2}{x} $$
$$ \sqrt{x} + 2\ln(x) $$
$$ \frac{2}{3}x^{3/2} + 2\ln(x) $$
$$ \sqrt{x} + \frac{2}{x} $$
Let $$K$$ be the function defined by $$ K(x) = \int_{0}^{x} \left( \sin(4t) + \cos(2t) \right) dt $$. What is $$ K'(x) $$?
$$ 4\cos(4x) - 2\sin(2x) $$
$$ -\frac{1}{4}\cos(4x) + \frac{1}{2}\sin(2x) $$
$$ \sin(4x) - \cos(2x) $$
$$ \sin(4x) + \cos(2x) $$
Let the function $$P$$ be defined by $$ P(x) = \int_{1}^{x} \left( \ln(t) + \frac{1}{t^3} \right) dt $$. Which of the following is an expression for $$ P'(x) $$?
$$ \ln(x) + \frac{1}{x^3} $$
$$ \ln(x) - \frac{1}{2x^2} $$
$$ \frac{1}{x} + \frac{3}{x^4} $$
$$ x\ln(x) - \frac{1}{2x^2} $$
Let the function $F$ be defined by $$F(x) = \int_{0}^{x} e^{2t} dt$$. What is $F'(x)$?
$2e^{2x}$
$e^{2x}$
$\frac{e^{2x}}{4}$
$\frac{e^{2x}}{2}$
If `g(x) = $\displaystyle\int_{2}$ˣ (t³ - 4t) dt`, then `g'(5) =`
$\displaystyle\int_{2}^{5}$ (t³-4t) dt
5³ - 4(5) = 105
(5³-4(5)) - (2³-4(2)) = 97
3(5)² - 4 = 71
If g(x) = $\displaystyle\int_{2}$ˣ sin(t²) dt, which of the following is g'(x)?
g'(x) = -sin(x²)
g'(x) = sin(2²)
g'(x) = 2x·cos(x²)
g'(x) = sin(x²)
Let the function `F` be defined by `F(x) = $\displaystyle\int_{1}$ˣ (t² - 3t + 2) dt`. Which of the following is the value of `F'(4)`?
-(4² - 3(4) + 2)
4² - 3(4) + 2
$\displaystyle\int_{1}^{4}$ (2t - 3) dt
4² - 3(4) + 2 - (1² - 3(1) + 2)
The graph above shows the function $$f(x)=2*x$$ on the interval $$[2,3]$$. Use the Fundamental Theorem of Calculus to compute the definite integral $$\int_{2}^{3} 2*x dx$$.
Approximately $$4$$
$$6$$
$$5$$
A misinterpretation might give $$7$$.
I.
$$\int_{2}^{3} 2*x dx = 3^2 - 2^2$$
II.
$$\int_{2}^{3} 2*x dx = (2*3) - (2*2)$$
III.
$$\int_{2}^{3} 2*x dx = \frac{3^2}{2} - \frac{2^2}{2}$$
Which of these statements correctly applies the Fundamental Theorem of Calculus to evaluate the integral $$\int_{2}^{3} 2*x dx$$?
I only
II only
III only
I and III
The area under the graph of $$f(x)=2x$$ on the interval $$[2, 5]$$ is given by the definite integral $$\int_{2}^{5} 2x dx$$. Which of the following correctly demonstrates the evaluation of this integral using the Fundamental Theorem of Calculus?
$$2*x^2 \Big|_{2}^{5}$$
$$x^2 \Big|_{2}^{5} + C$$
$$\frac{x^2}{2} \Big|_{2}^{5}$$
$$x^2 \Big|_{2}^{5} = 5^2 - 2^2$$
Use the Fundamental Theorem of Calculus to evaluate $$\int_{2}^{3} 2x dx$$.
$$3^2-2^2+C$$
$$\frac{2*(3^2)-2*(2^2)}{2}$$
$$5$$
$$3^2-2^2+1$$
All of the following statements about evaluating the definite integral $$\int_{2}^{3} 2x dx$$ are true EXCEPT:
An antiderivative of $$2*x$$ is $$x^2+C$$, and when evaluating a definite integral, the constant cancels out.
Using the Fundamental Theorem of Calculus, the integral is computed as $$F(2)-F(3)$$, where $$F(x)=x^2$$ is an antiderivative of $$2*x$$.
The definite integral is evaluated by computing the difference between the antiderivative at the upper limit and at the lower limit.
The correct application of the FTC gives $$\int_{2}^{3} 2*x dx = F(3)-F(2) = 9-4 = 5$$.
Which of the following expressions represents the evaluation of $$\int_{2}^{3} 2x dx$$ using the Fundamental Theorem of Calculus?
$$x^2 \Big|_{2}^{3}$$
$$\frac{x^2}{2} \Big|_{2}^{3}$$
$$2*x^2 \Big|_{2}^{3}$$
$$x^2 + C \Big|_{2}^{3}$$
If the function `h` is defined by h(x) = $\displaystyle\int_{1}$ˣ e^(t²)dt, which of the following is equal to h'(3)?
$\displaystyle\int_{1}^{3}$ e^(t²)dt
3e⁹
e⁹
2·3·e⁹
Let the functions f and g be defined by f(x) = $\displaystyle\int_{0}$ˣ e^(-t) dt and g(x) = $\displaystyle\int_{0}$ˣ f(t) dt. What is g'(x)?
e^(-x)
$\displaystyle\int_{0}$ˣ e^(-t) dt
x·e^(-x)
1 - e^(-x)
The Fundamental Theorem of Calculus establishes the relationship between which of the following concepts?
Differentiation and integration
Sequences and series
Vectors and matrices
Limits and continuity
Which of the following justifies evaluating a definite integral by finding the difference between the values of an antiderivative at the upper and lower limits of integration?
Fundamental Theorem of Calculus
Symmetry property
Integration by Parts
Linearity property
Which expression represents the evaluation of $$\int_{0}^{\pi/2} \cos(x) dx$$ using the Fundamental Theorem of Calculus?
$$\cos(\pi/2) - \cos(0)$$
$$\sin(\pi/2) + \sin(0)$$
$$\sin(\pi/2) - \sin(0)$$
$$\frac{\sin(\pi/2) - \sin(0)}{2}$$
The function $$G$$ is defined by $$ G(x) = \int_{2}^{x} \left( e^{2t} - 3t^2 \right) dt $$. Which of the following is an expression for $$ G'(x) $$?
$$ \frac{1}{2} e^{2x} - 3x^2 $$
$$ \frac{1}{2} e^{2x} - x^3 $$
$$ e^{2x} - x^3 $$
$$ e^{2x} - 3x^2 $$
The function $$ Q $$ is defined by $$ Q(x) = \int_{0}^{x} \left( \sec^2(3t) + 5t^4 \right) dt $$. What is $$ Q'(x) $$?
$$ \sec^2(3x) + 5x^4 $$
$$ 6\sec(3x)\tan(3x) + 20x^3 $$
$$ \frac{1}{3}\tan(3x) + x^5 $$
$$ \sec^2(x) + x^4 $$
Let the function $f$ be defined by $$f(x) = \int_{0}^{x} e^{-t} dt$$. What is the value of $f'(3)$?
e^(-3)
-e^(-3)
1 - e^(-3)
e^3
If the function $F$ is defined by $$F(x) = \int_{a}^{x} f(t) dt$$, then $F'(x) =$
F(x)
f(x)
0
f(t)
If the function $F$ is defined by $$F(x) = \int_{5}^{x}(2t+6)dt$$, then $F'(x) =$
$2$
$2x$
$2x+12$
$2x+6$
Which sequence correctly outlines the steps for evaluating a definite integral from `a` to `b` using the Fundamental Theorem of Calculus?
Find the antiderivative of the integrand, Evaluate at the upper limit, Evaluate at the lower limit, Subtract the lower evaluation from the upper evaluation
Evaluate at the lower limit, Find the antiderivative of the integrand, Subtract the lower evaluation from the upper limit, Evaluate at the upper limit
Find the antiderivative of the integrand, Subtract the lower evaluation from the upper evaluation, Evaluate at the lower limit, Evaluate at the upper limit
Subtract the lower evaluation from the upper evaluation, Find the antiderivative of the integrand, Evaluate at the upper limit, Evaluate at the lower limit
The derivative of a function $$f$$ is continuous on the closed interval $$[0, 4]$$. The table below shows selected values of $$f$$ and its derivative, $$f'$$.
| $$x$$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| $$f(x)$$ | 5 | 2 | 3 | 6 |
| $$f'(x)$$ | -3 | 1 | 3 | 4 |
Given that $$\displaystyle\int_{0}^{4} f'(t) dt = 8$$, what is the value of $$f(4)$$?
13
3
10
5
The function $$g$$ is continuously differentiable with $$g(1) = 6$$ and $$g'(1) = 3$$. What is the value of $$\displaystyle \lim_{x \to 1} \frac{\int_{1}^{x} g(t) dt}{g(x) - 6}$$?
0
2
1
$$\displaystyle \frac{1}{2}$$
All of the following statements about the Fundamental Theorem of Calculus are true EXCEPT:
The definite integral $$\int_{a}^{b} f(x) dx$$ is equal to $$F(a)-F(b)$$, where $$F(x)$$ is any antiderivative of $$f(x)$$.
If $$F'(x)=f(x)$$ for all x in [a, b] and $$f(x)$$ is continuous on [a, b], then $$\int_{a}^{b} f(x) dx = F(b)-F(a)$$.
For the Fundamental Theorem of Calculus to hold, the function $$f(x)$$ must be continuous on the closed interval [a, b].
Any two antiderivatives of a continuous function differ by a constant, so the constant cancels in the evaluation $$F(b)-F(a)$$.