Review for mid-term in MTH231

The mid-term will cover lessons 1-7 and 9-25 on the syllabus.

We have done a fair amount this semester:

Review

In lessons 1 and 2 we reviewed some concepts. The most important being

• the point-slope equation for a line $y = f(c) + m \cdot (x-c)$

• the definitions of $\sin(x)$, $\cos(x)$, $\tan(x)$; Further we have used the Pythagorean identity ($\sin^2(x) + \cos^2(x) = 1$) and the sum formulas ($\sin(x+h) = \sin(x)\cos(h) + \sin(h)\cos(x)$ and $\cos(x+h) = \cos(x)\cos(h) - \sin(x)\sin(h)$).

• the definition of an inverse function from a few viewpoints (algebraic, graphical).

• the main inverse function relation between exponential and logarithmic funcitons

In lessons 4-7, 9 we learned about limits and continuity. In particular we learned of:

• limits, left limits, right limits

• limit laws

• limits and continuity

• indeterminate forms

• some basic tricks to find limits when a limit is indeterminate: canceling factors; multipliying by a conjugate; combining rational functions to cancel.

Continuity (that $\lim_{x \rightarrow c} f(x) = f(c)$) is a key concept and can be interpreted as $f(x + h) - f(x)$ goes to $0$ as $h$ goes to $0$.

The intermediate value theorem is a consequence of continuity and important for theoretical reasons. However, it will not make an appearance on the exam.

The lessons 10-23 are on derivatives. We start with the definition in terms of a limit, but then move onto rules for computing derivatives. These rules allow the computation of a derivative without needing to take a limit.

The definition is in terms of a point ($c$): the derivative is the limit of the slope of the secant line between $c$ and $c+h$ as $h$ goes to $0$. This is quickly turned into the derivative as a function of $x$.

The rules to find derivatives are many but still limited:

• sum/difference and constant multiples

• product rule

• quotient rule

• chain rule

• power rule

• $\sin$ and $\cos$ rules

• $e^x$ and $\log(x)$ rules

Implicit differentiation is a technique that allows one to find the slopes of tangent lines to equations (not just functions). The basic insight is to use y is secretly a function of x so the chain rule must be used with expressions containing y.

Calculus has a large number of standard applications of the derivative. For this test, we will have a question on related rates. At one level this is nothing more than going from an equation relating two quantities to a new equation relating rates of change for the same two quantities. The hardest part is usually going from the problem to a picture to an equation.

Sample problems

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. (If something seems wrong, best to ask. There is a always a chance I made a mistake ;)

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Review

• 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:

$4\pi$

$\pi/4$

$7\pi/4$

$4\pi/7$

• The sine 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

Limits

• 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

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

5×2 Array{Float64, 2}:
 0.1     2.2140275816016985
 0.01    2.0201340026755776
 0.001   2.002001334000303
 0.0001  2.000200013334563
 1.0e-5  2.0000200001257795

• 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 $\lim_{x \rightarrow 1-} f(x)$

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

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

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

• 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, and $f$ is continuous everywhere. 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 a common factor

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 a common factor

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 a common factor

multiply by the conjugate

use continuity

• Evaluate the limit when $a$ is a constant

$$~ \lim_{h \rightarrow 0} \frac{1/(a+h) - 1/a}{h} ~$$

$1/a$

$-1/a$

$1/a^2$

$-1/a^2$

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

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

• 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 functions is not continuous at $0$?

$\sin(x)$

$\tan(x)$

$e^x$

$\sqrt{x}$

Derivatives

• 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?

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

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

$(0, 4)$

• Which rule would you use first to find the derivative of $f(x)$ when

$$~ f(x) = x \cdot (1 + \sin(x)) ~$$

constant multiple

sum/difference

product

quotient

chain

• Which rule would you use first to find the derivative of $f(x)$ when

$$~ f(x) = (1 + \sin(x))^{1/3} ~$$

constant multiple

sum/difference

product

quotient

chain

• Which rule would you use first to find the derivative of $f(x)$ when

$$~ \frac{1 + x^2}{1 - x^2} ~$$

constant multiple

sum/difference

product

quotient

chain

• What is the outer function (for the chain rule)

$$~ f(x) = \sin^2(x^2 - 1) ~$$

$x^2$

$\sin(x)$

$x^2 - 1$

• What is the outer function (for the chain rule)

$$~ f(x) = e^{-\frac{1}{2}(x-1)^2} ~$$

$e^x$

$(1/2)(x^2 - 1)$

$x^2$

• 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 $f(x) = (\sqrt{x}+1)\cdot x^2$.

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

• Using rules of derivative, find $f'(2)$ when $f(x) = \log(x + \sin(x))$.

• The following figure shows the graph of an ellipse $x^2/4 + y^2 = 1$.

using ImplicitEquations, Roots
a = 1
f(x,y) = x^2/4 + y^2
plot(Eq(f,1))

At a point $(x,y)$ what is $dy/dx$?

$-x/(4y)$

$4x/y$

-4x/y

$x/(4y)$

• The length of a rectangle is increasing at the rate of 4 ft/hr. and the width of the rectangle is decreasing at the rate of 3 ft/hr. At what rate is the rectangle's perimeter changing when the length is $8$ and the width $5$?

At the same point, what is the rate of change of area?

(For both, assume both $w$ and $l$ are functions of time.