Review for Test 1 in MTH231

We will cover lessons 1-13 (through 3.3)

Review for Test 1 MTH231 Professor Verzani

These are sample problems, they need not be exhaustive or representative of what will be on the exam, but should help prepare you. They are actually on the easier side, as many can be done "in your head." Please also review examples done in class, the examples in the book and the homework questions in your preparation for this exam.

Chapter 1

• Is $3$ in the interval $[3, 4)$?

Yes

No

is $4$?

Yes

No

• is the distance between $(3,4)$ and $(6,8)$ equal to $25$?

Yes

No

• Is $|2x - 3| < 4$ equivalent to $-1 < 2x < 7$?

Yes

No

• The graph of the function $h(x) = f(x + c) + d$ is the same as the graph of $f(x)$ only

Shifted up by c and right by d

Shifted up by d and right by c

Shifted up by c and left by d

Shifted up by d and left by c

• The graph shows a function $f$ in red and $g$ in blue. The graph of $g$ is the graph of $f$ with what transformation

shifted right by 2 and up by 2

shifted right by 2 and up by 4

shifted left by 2 and up by 2

shifted left by 2 and up by 4

• A function is increasing if for all $x < y$ it must be that $f(x) < f(y)$.

Yes

No

• The function $f(x) = x^{11}$ is an odd function

Yes

No

• A line goes through the point $(2,4)$ with slope $8$. An equation for the line is $y = 2 + 4\cdot(x-8)$.

Yes

No

• A line goes through the point $(0, 10)$ and has slope $-3$. An equation for the line is $y = -3x + 10$.

Yes

No

• Is the equation of the line $x = 3$ for a horizontal line or vertical line?

horizontal

vertical

• If $p(x) = 4x^2 - 1x + 3$ does $p(x) = 4(x-3/2)^2 -6$ complete the square?

Yes

No

• If $p(x) = x^3 - 2x - 3$, what is $p(0)$?

• If $p(x) = -3(x - 2) + 1$, the coordinates of the vertex are:

$(-3, -2)$

$(-2, 1)$

$(-3, 2)$

$(2, 1)$

• We can construct new functions from $f$ and $g$ by any of $c\cdot f(x)$, $f(x) + g(x)$, $f(x) \cdot g(x)$, $f(x)/g(x)$, and $f(g(x))$.

Yes

No

• The points $(1,2)$, $(3,4)$, and $(5, 6)$ are on the graph of $f(x)$ and the points $(2,1)$, $(4,2)$, and $(6,3)$ are on $g(x)$. What is $f(g(6))$?

• What degree measurement is $\pi/3$?

• Which of these is an angle in quadrant 4:

$4\pi$

$\pi/4$

$7\pi/4$

$4\pi/7$

• Thesine of $5\pi/6$ is which?

$0$

$1/2$

$\sqrt{2}/2$

$\sqrt{3}/2$

$1$

• Characterize the following graph

polynomial

sinusoidal

exponential decay

exponential growth

• Characterize the following graph

polynomial

sinusoidal

exponential decay

exponential growth

• Which of these functions is an odd function?

$\sqrt{x}$

$e^x$

$\cos(x)$

$\sin(x)$

• The domain of arcsine is:

$[-1,1]$

$[-\pi/2, \pi/2]$

$[0, \pi]$

• The range of arccos is:

$[-1,1]$

$[-\pi/2, \pi/2]$

$[0, \pi]$

• What is $\log_6(9) + \log_6(4)$?

• Which is bigger?

$\log_{10}(100)$

$\log_5(100)$

$\log_e(100)$

• Suppose $b>1$, is $b^{(x+y)/2} = \sqrt{b^x + b^y}$?

Yes

No

Chapter 2

• Consider the following graph

The average rate of change over $[1,2]$ is

positive

negative

zero

The average rate of change over $[2,3]$ is

positive

negative

zero

The average rate of change over $[0, 2]$ is

positive

negative

zero

The instantaneous rate of change at $2$ is

positive

negative

zero

The instantaneous rate of change at $3$ is

positive

negative

zero

• Consider the graph below over the interval $[0,2]$. Graphically identify a value $c$ with instantaneous velocity equal to the average velocity over $[0,2]$.

• The following table suggests what limiting value for

$$\lim_{h\rightarrow 0}((e^{2h}-1)/h)$$

• From the graph, estimate $\lim_{t \rightarrow 0} (1+r)^(1/r)$.

f(r) = (1 + r)^(1/r)
plot(f, -1/2, 1)

• From the graph answer the following (large points indicate the function is defined at a point):

Find $\lim_{x -> 1-} f(x)$

Find $\lim_{x -> 1+} f(x)$

Find $\lim_{x -> 2+} f(x)$

Find $\lim_{x -> 4-} f(x)$

• Which function below has only a one-sided limit at $x=0$?

$\sin(x)$

$e^x$

$\sqrt{x}$

$\tan(x)$

• Does this graph appear to have a limit at $x=0$?

Yes

No

• Given $\lim_{x\rightarrow c}f(x)=L$, $\lim_{x\rightarrow c}g(x)=M$, $\lim_{x\rightarrow M}f(x)=N$, $\lim_{x\rightarrow L}g(x)=O$, all non-zero, compute:

The value of $\lim_{x \rightarrow c}(f(x)^2 + 2f(x)g(x) + g(x)^2)$.

$L + 2ML + M$

$L^2 + 2ML + M^2$

$N^2 + 2NO + O^2$

does not exist

The value of $\lim{x \rightarrow c}(f(x)/f(g(x))$

$L/M$

$L/N$

$L/O$

Does not exist

The value of $\lim_{x \rightarrow c}(10f(x)^3 + 5f(x)^2 + x)$

$10L^3 + 5L^2 + L$

$10L^3 + 5L^2 + c$

$10c^3 + 5c^2 + c$

• Is the function $f(x) = \sqrt{x^2 + 9}$ always continuous?

Yes

No

• Does the function $f(x) = (x^2 - 4)/(x-2)$ have a removable discontinuity?

Yes

No

• Does the function $f(x) = |x|/x$ have a removable discontinuity?

Yes

No

• For the following functions, which has a right limit at $c=0$ which can be found simply by "plugging" in? (That is evaluating at $0$ is not indeterminate)?

$x^x$

$\sin(x)/x$

$(\cos(x)-1)/x$

$\cos(x)/(x-1)$

$x\cdot\log(x)$

• What "trick" allows you to algebraically find the limit at $0$ of $(x^2-4x+3)/(x^2+x-12)$?

transform algebraically and cancel

multiply by the conjugate

use continuity

What "trick" allows you to algebraically find the limit at $0$ of $\cos(x)/(x-1)$?

transform algebraically and cancel

multiply by the conjugate

use continuity

What "trick" allows you to algebraically find the limit at $0$ of $(\sqrt{x}-3)/(x-9)$?

transform algebraically and cancel

multiply by the conjugate

use continuity

• Evaluate the limit when $a$ is a constant

$$~ \lim_{x \rightarrow a} \frac{1/h - 1/a}{h-a} ~$$

$1/a$

$-1/a$

$1/a^2$

$-1/a^2$

something else

• Evaluate the limit:

$$~ \lim{x \rightarrow 0} \frac{\sin(x)}{1 + \cos(x)} \cdot \frac{\sin(x)}{x}. ~$$

• Evaluate the limit

$$~ \lim{x \rightarrow 0} \frac{\sin(9x)}{3x} ~$$

• Evaluate the limit

$$~ \lim{x \rightarrow 0} \frac{1 - \cos(10^9 \cdot x)}{x} ~$$

• Evaluate the limit

$$~ \lim{x \rightarrow 0} \frac{\sin(x)^2}{x} ~$$

• Evaluate the limit

$$~ \lim{x \rightarrow 0} \frac{\sin(\sin(x))}{x} ~$$

• You are given $4x - x^2 \leq f(x) \leq x^2 + 2$. What is the limit

of $f$ at $c=1$?

0

1

2

3

can't tell

• Which of these has a limit of $0$ as $x$ "goes to" infinity?

$(x^2 - 2x + 3)/(x^2 - 3x + 2)$

$(x^{1000} - 1000x^1) / (x^{1001} - 1001 x^1)$

$(x^a - 1)/(x^{a-1} + 1)$

• Which of these has a finite, non-zero limit as $x$ "goes to" infinity?

$(x^2 - 2x + 3)/(x^2 - 3x + 2)$

$(x^{1000} - 1000x^1) / (x^{1001} - 1001 x^1)$

$(x^a - 1)/(x^{a-1} + 1)$

• Is the limit as $x$ goes to infinity and as $x$ goes to minus

infinity the same for $f(x) = (12x+25)/\sqrt{16x^2 + 4}$?

Yes

No

• What is

$$~ \lim_{x \rightarrow \infty}\frac{3x^{7/2} + 7x^{3/2}}{x^2 - x^{1/2}}? ~$$

infinity

$3/1$

$7/2 -2$

• Which of these functions is not continuous at $0$?

$\sin(x)$

$\tan(x)$

$e^x$

$\sqrt{x}$

• Let $m(b) = \lim_{h\rightarrow 0}(b^h-1)/h$. If $m(b)$ is continuous, increasing, and $m(1) < 1$ and $m(3) > 1$ why is there some value with $m(x) = 1$ in the interval $[1,3]$?

The intermediate value theorem (IVT)

because the limit at $x$ goes to $1$ exists

because the limit at $x$ goes to $3$ exists

• Does the IVT theorem guarantee a solution to $f(x) = 0$ in the interval $[-1,1]$ when $f(x)=1/x$?

Yes

No, as $f(-1)$ has the same sign as $f(1)$

No, as $f(x)$ is not continuous on $(0, 1)$

No, as $f(x)$ is not continuous on $[-1, 1]$

Chapter 3 (through 3.3)

• For a function $f$, the expression $(f(a+h) - f(a))/h$ is

The derivative of $f$ at $x=a$.

The slope of the tangent line of $f$ at $x=a$ and $x=h$.

The slope of the secant line between $(a,f(a))$ and $(a+h, f(a+h))$.

• Let $f(x) = 1/x$. The expression $(f(a+h) - f(a))/h$ is

$(1/a + h - 1/a)/h$

$(1/a + h) - (1/a))/h$

$(1/(a + h) - 1/a)/ h$

• Let $f(x) = 12x^2 + 8x$. What is $f(a + b)$?

$12a^2 + 8b$

$12a^2 + b + 8a$

$12(a+b)^2 + 8(a+b)$

• From this plot

Which line is a tangent line?

The red one

the blue one

neither

• From this plot

Estimate the value of $f'(1)$.

• From this plot

On which intervals is the derivative positive?

$(1,2) \text{ and } (3,4)$

$(0, 1) \text{ and } (2, 3)$

$(0, 4)$

• Using the definition of the limit, compute the derivative of $f(x)=3x^2$.

$3x^2$

$2x^3$

$6x^2$

$6x$

• Using rules of derivatives, find $f'(2)$ when $f(x) = 16x^2 + 32x$.

• Using rules of derivatives, find $f'(2)$ when $f(x) = x^2 \sin(x)$.

• Using rules of derivatives, find $f'(2)$ when $f(x) = x*e^x$.

• Using rules of derivatives, find $f'(2)$ when $f(x) = (x+3)/(x^2 + 2x + 4)$.

• Using rules of derivatives, find $f'(2)$ when $f(x) = (\sqrt{x}+1)\cdot x^2$.