2.) If $f(x)$ is continuous on a closed interval, then it is enough to
look at the points where $f'(x) = 0$ in order to find its absolute maxima
and minima.
A) True
B) False
3.) If $f''(a) = 0$, then $f$ has an inflection point at $a$.
A) True
B) False
1.) For extremely large positive $x$,
$x^{255} < (1.01)^x$
A) True
B) False
2.) For large positive $x$,
$x^{255} < (.99)^x$
A) True
B) False
3.) For large positive $x$,
$2^x < e^x< 3^x$
A) True
B) False
1.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4\sqrt{x}}$,
then $f$ is an increasing function on $(0, \infty)$.
A) True
B) False
2.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4ln(x)
\sqrt{x}}$, then $f$ is an increasing function on $(0, \infty)$.
A) True
B) False
3.)
If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4ln(x^2)
\sqrt{x}}$, then $f$ is an increasing function on $(0, \infty)$.
A) True
B) False
1.) A number $M$ exists such that $ln(x) \leq M$ for all $x > 0$.
A) True
B) False
2.) There exists a function $f$ such that the derivative of $f$ =
$f'(x) = {e^x(x^2 + 1) \over 4ln(x) \sqrt{-x}}$.
A) True
B) False
1.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4\sqrt{x}}$,
then $f$ is an increasing function on $(0, \infty)$.
A) True
B) False
2.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4\sqrt{-x}}$,
then $f$ is an increasing function on $(-\infty, 0)$.
A) True
B) False
3.) If the derivative of $f$ = $f'(x) = {\sqrt{x^2+3} \over e^x(x^2 +
1)}$,
then $f$ is an increasing function.
A) True
B) False
1.) If $f$ is differentiable, then its derivative is unique.
A) True
B) False
2.) The anti-derivative of a function is unique.
A) True
B) False
3.) The anti-derivatives of $f'$ are the
functions $f + C$, where $C$ is a constant.
A) True
B) False
4.) The unique derivative of a differentiable function $f$ is the
function $f'$.
1.) If $f$ is continuous, then
$\int_2^9 f(x)dx$ is a number
A) True
B) False
2.)
If $f$ is continuous, then $\int f(x)dx$ is a number
A) True
B) False
3.)
If $f$ is continuous, then $\int f(x)dx$ is a collection of functions of
the form $F(x) + C$ where $F'(x) = f(x)$.
A) True
B) False
4.)
If $f$ is continuous, then $\int f(x)dx$ is a collection of functions of
the form $F(x) + C$ where $f'(x) = F(x)$.
A) True
B) False
5.)
If $f$ is continuous for all real numbers, then there is a unique
function $F$ that satisfies
(i) $F'(x) = f(x)$ and (ii) $F(3) = 4$.
A) True
B) False
6.)
$\int_{-2}^2 x^4 dx = 0$
A) True
B) False
7.)
$\int_{-2}^2 x^3 dx =
2\int_{0}^2 x^3 dx$
A) True
B) False
8.) $\int_{-3}^4 e^x dx < 0$
A) True
B) False
9.) $\int_{-3}^4 e^{-x} dx > 0$
A) True
B) False
10.) $\int_{-3}^4 ln(x) dx < 0$
A) True
B) False
11.) $\int_{1}^4 ln(x) dx < 0$
A) True
B) False
12.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4\sqrt{x}}$,
then $f$ is a decreasing function on $(0, \infty)$.
A) True
B) False
13.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4ln(x)
\sqrt{x}}$, then $f$ is an increasing function on $(0, \infty)$.
A) True
B) False
14.)
If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4ln(x^2)
\sqrt{x}}$, then $f$ is an increasing function on (0, 1).
A) True
B) False
15.) If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4ln(x)
\sqrt{x}}$, then $f$ is an increasing function on $(1, \infty)$.
A) True
B) False
16.)
If the derivative of $f$ = $f'(x) = {e^x(x^2 + 1) \over 4ln(x^2)
\sqrt{x}}$, then $f$ is an decreasing function on (0, 1).
A) True
B) False
1.) The decay of a radioactive substance is modeled by a differential
equation.
A) True
B) False
2.) One of the fundamental questions concerning any initial value problem
is as follows: Does the problem have a solution?
A) True
B) False
3.) One of the fundamental questions concerning any initial value problem
is as follows: Does the problem have more than one solution?
A) True
B) False
1.) Numerical approximations for solutions to differential equations are
often needed as
the solutions to many differential equations cannot be expressed
algebraically.
A) True
B) False
2.) If a computer is used to find a numerical approximation to a
differential equation,
then we know the equation has at least one solution.
A) True
B) False
1.) If $f$ is continuous, then on an appropriate domain, the initial
value problem $y' = f'(t)$, $y(t_0) = y_0$ has a unique solution.
A) True
B) False
2.) If $f$ is continuous, then on an appropriate domain, the initial
value problem $y' = f'(t, y)$, $y(t_0) = y_0$ always has a unique
solution.
A) True
B) False
3.) The initial value problem $y' = y^\frac{1}{3}$, $y(2) = 0$ has three
different solutions.
A) True
B) False
4.) The initial
value problem $y' = f'(t, y)$, $y(t_0) = y_0$ might not have a
solution.