Let $F$ be the function defined by $$F(x) = \int_{1}^{x} (t^3 - 4t^2 + 2) dt$$. If $G(x) = F(x^2)$, what is the value of $G'(2)$?
The second part of the FTC: differentiating integrals, variable limits, and the chain rule.
40 questions
AP Learning Target(s): LT 4: The Fundamental Theorem of Calculus and Accumulation Functions
Let $F$ be the function defined by $$F(x) = \int_{1}^{x} (t^3 - 4t^2 + 2) dt$$. If $G(x) = F(x^2)$, what is the value of $G'(2)$?
28
8
14
112
$$Let the function `F` be defined by `F(x) = \int_{0}^{x} f(t) dt`. Which of the following represents `F'(x)`?$$
The slope of `f(t)` at `x`
The area under `f(t)` from `0` to `x`
The average value of `f(t)`
The value of `f(x)`
Let the function $$ M $$ be defined by $$ M(x) = \int_{2}^{x} \left( 3t^2 - \frac{4}{t^2} \right) dt $$. What is $$ M'(x) $$?
$$ 6x + \frac{8}{x^3} $$
$$ x^3 + \frac{4}{x} $$
$$ 3x^2 - \frac{4}{x^2} $$
$$ x^2 - \frac{1}{x^2} $$
Let the function `F` be defined by `F(x) = $\displaystyle\int_{0}$ˣ (t³ - 4t) dt`. The function `F` is an antiderivative of which of the following functions?
f(x) = (x⁴/4)-2x²
f(x) = F'(x)
f(x) = F(x) + C for some constant C
f(x) = x³-4x
Let F be a function defined by F(x) = $\displaystyle\int_{1}$ˣ √(1+t³) dt. If the function G is defined by G(x) = F(x²), what is the value of G'(2)?
2√(1+16)
4√(1+64)
√(1+64)
4√(1+16)
Let the function `g` be defined by `g(x) = $\displaystyle\int_{2}$ˣ f(t)dt`, where `f` is a continuous function. If `g'(3) = 5` and `g''(3) = -2`, what are the values of `f(3)` and `f'(3)`?
f(3) = -5, f'(3) = -2
f(3) = -2, f'(3) = 5
f(3) = 5, f'(3) = -2
f(3) = 5, f'(3) = 2
If $$F(x) = \int_{x}^{x^2} \frac{\sin(t)}{t} dt$$, what is $F'(x)$?
$\frac{4x^2\sin{x^2}}{x^2}-\frac{\sin(x)}{x}$
$\frac{2x\sin{x^2}}{x^2}-\frac{\sin(x)}{x}$
$\frac{4x^2\sin{x^2}}{x}-\frac{\sin(x)}{x}$
$\frac{\sin{x^2}}{x^2}-\frac{\sin(x)}{x}$
Let the function $$F$$ be defined by $$F(x) = \displaystyle\int_{4}^{x^2} \sqrt{t} dt$$ for $$x > 0$$. Which of the following is $$F'(x)$$?
$$\displaystyle\dfrac{2x^3 - 16}{3}$$
$$\displaystyle-\dfrac{1}{2x}$$
$$\displaystyle 2x^2$$
$$\displaystyle\sqrt{x}$$
Let the function $$f$$ be defined by $$f(x) = \int_{0}^{x^3} \cos(t^2) dt$$. What is the value of $$f'(\sqrt{\pi})$$?
$$\cos(\pi^3)$$
$$3\pi \cos \pi$$
$$3\pi \cos(\pi^3)$$
$$3\pi \sin(\pi^3)$$
$$\displaystyle \frac{d}{dx} \int_{e}^{x^3} \ln(t^2 + 1) dt =$$
$$\ln(x^6 + 1) - \ln(e^2 + 1)$$
$$\ln(x^6 + 1)$$
$$3x^2 \ln(x^6 + 1)$$
$$3x^2 \ln(x^2 + 1)$$
If the function $$J$$ is defined by $$J(x) = \displaystyle\int_{0}^{\ln x} \cos(e^t) dt$$, what is the value of $$J'(e)$$?
$$\displaystyle \frac{\cos(1)}{e}$$
$$\displaystyle \frac{\cos(e)}{e}$$
$$\displaystyle e \cos(e)$$
$$\displaystyle \cos(1)$$
Let the function $$K$$ be defined by $$K(x) = \displaystyle\int_{0}^{\sin x} e^{t^2} dt$$. What is the value of $$K'\left(\frac{\pi}{6}\right)$$?
$$\displaystyle e^{\frac{1}{4}}$$
$$\displaystyle \frac{\sqrt{3}}{2} e^{\frac{1}{4}}$$
$$\displaystyle \frac{\sqrt{3}}{4} e^{\frac{1}{2}}$$
$$\displaystyle \frac{1}{2} e^{\frac{\sqrt{3}}{2}}$$
Let the function $$g$$ be defined by $$g(x) = \displaystyle\int_{1}^{x^2} \cos(t^3) dt$$. What is the value of $$g'(\sqrt{2})$$?
$$\displaystyle 4 \cos(2)$$
$$\displaystyle \sqrt{2} \cos(8)$$
$$\displaystyle 2\sqrt{2} \cos(8)$$
$$\displaystyle 2 \cos(4)$$
Let $$N(x) = \displaystyle\int_{\sqrt{x}}^{x^2} \sec(t^3) dt$$. What is the value of $$N'(1)$$?
$$\displaystyle \frac{5}{2}\sec(1)$$
$$\displaystyle \frac{3}{2}\sec(1)$$
$$\displaystyle \frac{1}{2}\sec(1)$$
$$\displaystyle 2\sec(1)$$
Let the function $$F$$ be defined by $$F(x) = \displaystyle\int_{x^2}^{4} e^{t^2} dt$$. What is the value of $$F'(1)$$?
$$\displaystyle -e$$
$$\displaystyle -2e$$
$$\displaystyle 2e$$
$$\displaystyle -4e$$
Let $$G$$ be the function defined by $$G(x) = \displaystyle\int_{x}^{x^3} \ln(t+1) dt$$. What is the value of $$G'(2)$$?
$$\displaystyle 12\ln(9) + \ln(3)$$
$$\displaystyle 11\ln(3)$$
$$\displaystyle 24\ln(3)$$
$$\displaystyle 12\ln(9) - \ln(3)$$
Let the function $$g$$ be defined by $$g(x) = \int_{2}^{x^3} \sin(t^2) dt$$. What is the value of $$g'(4)$$?
$$16\sin(64)$$
$$12\sin(4096)$$
$$48\sin(4096)$$
$$3\sin(64)$$
If the function `h` is defined by `h(x) = $$\int$$(x to x²) ln(t) dt`, then `h'(x)` is equal to which of the following?
h'(x) = ln(x²) - ln(x)
h'(x) = ln(x²) - ln(x) - 1
h'(x) = ln(x²) + ln(x)
h'(x) = 2x·ln(x²) - ln(x)
If $$F(x) = \int_{2}^{\cos^2(x)}(3t-1)dt$$, what is $F'(x)$?
$-2\sin(x)\cos(x)(3\cos^2(x)-1)$
$2\sin(x)\cos(x)(3\cos^2(x)-1)$
$-2\sin(x)\cos(x)(3\sin^2(x)-1)$
$2\sin(x)\cos(x)(3\sin^2(x)-1)$
Let the function $$P$$ be defined as $$P(x) = \displaystyle\int_{x^3}^{2x} \frac{1}{1+t^2} dt$$. What is the value of $$P'(1)$$?
$$\displaystyle \frac{1}{2}$$
$$\displaystyle -\frac{1}{10}$$
$$\displaystyle \frac{11}{10}$$
$$\displaystyle -\frac{11}{10}$$
If $$h(x) = \displaystyle\int_{0}^{x^2} \sin(t^2) dt$$, what is the value of $$h'(2)$$?
$$\displaystyle 8 \sin(4)$$
$$\displaystyle 4 \sin(16)$$
$$\displaystyle 2 \sin(4)$$
$$\displaystyle \sin(16)$$
If $$\displaystyle x > 0$$, then $$\displaystyle \frac{d}{dx} \int_{1}^{\sqrt{x}} \frac{1}{1+t^2} dt =$$
$$\displaystyle \frac{\sqrt{x}}{1+x}$$
$$\displaystyle \frac{1}{2\sqrt{x}(1+\sqrt{x})}$$
$$\displaystyle \frac{1}{1+x}$$
$$\displaystyle \frac{1}{2\sqrt{x}(1+x)}$$
The function $$f$$ is defined by $$f(x) = \displaystyle \int_{4}^{2x} \sqrt{t^2 - t} dt$$. What is the value of $$f'(2)$$?
$$\displaystyle\sqrt{2}$$
$$\displaystyle 0$$
$$\displaystyle 2\sqrt{12}$$
$$\displaystyle\frac{7}{2\sqrt{12}}$$
Let a function be defined by f(x) = $\displaystyle\int_{1}^{x²}$ t³ dt. What is f'(x)?
3x²
2x³
2xt²
x³
If $$F(x) = \int_{0}^{x^2} \frac{1}{\sqrt{t}} dt$$, what is $F'(x)$?
$\frac{x}{2}$
$\frac{1}{x}$
$\frac{1}{\sqrt{x}}$
$2$
Let F be the function defined by F(x) = $\displaystyle\int_{1}$ˣ √(1+t³) dt. What is the value of F''(2)?
0
3·2²·(1+2³)^(-1/2)
√(1+2³)
(3/2)·2²·(1+2³)^(-1/2)
Let F be the function defined by F(x) = $\displaystyle\int_{1}$^x √(1+t³) dt. What is the value of F''(2)?
√(1+2³)
3(2²)
0
3(2²)/(2√(1+2³))
Let $F$ be the function defined by $$F(x) = \int_{2x}^{4}\sin(t)dt$$. What is $F'(x)$?
$-2\sin(2x)$
$-2\sin(x)$
$2\sin(x)$
$2\sin(2x)$
If $$F(x) = \int_{2}^{x^3}(2t^2+t)dt$$, then $F'(x) =$
$6x^8+3x^5$
$2x^5+3x^2$
$2x^2+x$
$6x^3+2x$
Let the function $F$ be defined by $$F(x) = \int_{1}^{x^2} \frac{1}{t} dt$$. What is $F'(x)$?
$\frac{1}{x}$
$\frac{2}{x^2}$
$\frac{1}{2x}$
$\frac{2x}{x^2}$
Let the function $$M$$ be defined by $$M(x) = \displaystyle\int_{x}^{e^x} \tan^{-1}(t^2) dt$$. What is the value of $$M'(0)$$?
$$\displaystyle \frac{\pi}{2}$$
$$\displaystyle 1 - \frac{\pi}{4}$$
$$\displaystyle \tan^{-1}(2)$$
$$\displaystyle \frac{\pi}{4}$$
Let $$H(x) = \displaystyle\int_{x^2}^{2x} \sqrt{t^3 + 1} dt$$. What is the value of $$H'(1)$$?
$$\displaystyle 4 - \sqrt{2}$$
$$\displaystyle 6 + 2\sqrt{2}$$
$$\displaystyle 2\sqrt{9} - \sqrt{2}$$
$$\displaystyle 6 - 2\sqrt{2}$$
Let the function $$R$$ be defined by $$ R(x) = \int_{3}^{x} \left( \frac{2}{\sqrt{t}} + t^{3/2} \right) dt $$. Which of the following is an expression for $$R'(x)$$?
$$ -\frac{1}{x^{3/2}} + \frac{3}{2}\sqrt{x} $$
$$ \frac{1}{\sqrt{x}} + \sqrt{x^3} $$
$$ \frac{2}{\sqrt{x}} + x^{3/2} $$
$$ 4\sqrt{x} + \frac{2}{5}x^{5/2} $$
Which of the following equations represents the Second Fundamental Theorem of Calculus?
$$\int_{a}^{a}f(x) dx = 0$$
$$\frac{d}{dx}\int_{a}^{x}f(t)dt=f(x)$$
$$\int_{a}^{b}f(t) dt = F(b)-F(a)$$
$$\int_{a}^{b} f(x) dx = \lim\limits_{n \to \infty} \sum\limits_{i=1}^{n}\Delta x \cdot f(x_i)$$
Let the function `h` be defined by `h(x) = $\displaystyle\int_{0}$ˣ e^(t³) dt`. If `h(a) = 5` for some constant `a`, which of the following statements must be true?
h'(a) = e^(a³)
a = 5
h'(a) = 3a²e^(a³)
$\displaystyle\int_{0}$ᵃ e^(t³) dt = 0
Which of the following statements correctly defines the Second Fundamental Theorem of Calculus?
If f(x) is differentiable on an interval [a, b], then the integral of f'(x) is equal to F(x) evaluated at b minus F(x) evaluated at a
If f(x) is differentiable on an interval [a, b], then the integral of f'(x) is equal to f(b) - f(a)
If f(x) is continuous on an interval [a, b], then the integral of f(x) is equal to F(b) - F(a), where F(x) is the antiderivative of f(x)
If f(x) is continuous on an interval [a, b], then the integral of f(x) is equal to the derivative of F(x) evaluated at b
Let g(x) = $\displaystyle\int_{0}$ˣ sin(t²)dt. What is the value of g'(π)?
cos(π²)
sin(π²)
2π·cos(π²)
$\displaystyle\int_{0}$ᵖ 2t·cos(t²)dt
The Second Fundamental Theorem of Calculus establishes a relationship between which of the following concepts?
Algebra and Calculus
Accumulation and Decumulation
Differentiation and Integration
Chain Rule and Reverse Chain Rule
If $$F(x) = \int_{0}^{x} (1 + \cos(t)) dt$$, what is $F'(x)$?
$\cos(x)$
$1 + \cos(x)$
$-\sin(x)$
$\sin(x)$
Let g be the function defined by g(x) = $$\int$$(1 to x) sin(t²) dt. What is the value of g'(4)?
sin(16)
sin(4²)
$$\int$$(1 to 4) cos(t²)·2t dt
sin(1) - sin(4)