5. the difference between an average rate of change and an instantaneous rate of change is that
a) an average rate of change is always slower than an instantaneous rate of change.
b) an average rate of change represents the slope of a tangent line and an instantaneous rate of change represents the slope of a secant line.
c) an average rate of change is calculated over an interval and an instantaneous rate of change is calculated at one point.
d) an average rate of change can have a value of 0 but an instantaneous rate of change cannot.

6. a wedding caterer charges a $200 service fee and $24 per plate (cost of food for one guest). johanna has gotten rsvps accepting the invitation from 72 guests, but is still waiting to hear back from the other 18. she also has to pay for her own food and the groom’s food, who are not considered guests. let ( c(p) ) represent johanna’s catering cost for ( p ) plates of food. what is the range of ( c )?
a) ( 1976 leq c(p) leq 2408 )
b) ( 74 leq c(p) leq 92 )
c) ( 1776 leq c(p) leq 2208 )
d) ( 1928 leq c(p) leq 2360 )

7. selected values of a quadratic function ( h ) are given in the table. find the value of ( j + k ).

| ( x ) | 13 | 15 | 17 | 19 | 21 | 23 |
| ( h(x) ) | ( j ) | ( -8 ) | ( -3 ) | ( -1 ) | ( -2 ) | ( k ) |

a) ( -22 )
b) ( -20 )
c) ( -19 )
d) ( -6 )

8. the graph of a smooth, continuous function ( f ) has a single inflection point at ( x = 3 ). which of the following statements is true?
a) ( f ) is increasing at ( x = 3 )
b) the graph of ( f ) is concave down at ( x = 3 )
c) ( f ) has an absolute minimum value at ( x = 3 )
d) the rate of change of ( f ) changes from increasing to decreasing or decreasing to increasing at ( x = 3 )

Question

5. the difference between an average rate of change and an instantaneous rate of change is that
a) an average rate of change is always slower than an instantaneous rate of change.
b) an average rate of change represents the slope of a tangent line and an instantaneous rate of change represents the slope of a secant line.
c) an average rate of change is calculated over an interval and an instantaneous rate of change is calculated at one point.
d) an average rate of change can have a value of 0 but an instantaneous rate of change cannot.

6. a wedding caterer charges a $200 service fee and $24 per plate (cost of food for one guest). johanna has gotten rsvps accepting the invitation from 72 guests, but is still waiting to hear back from the other 18. she also has to pay for her own food and the groom’s food, who are not considered guests. let ( c(p) ) represent johanna’s catering cost for ( p ) plates of food. what is the range of ( c )?
a) ( 1976 leq c(p) leq 2408 )
b) ( 74 leq c(p) leq 92 )
c) ( 1776 leq c(p) leq 2208 )
d) ( 1928 leq c(p) leq 2360 )

7. selected values of a quadratic function ( h ) are given in the table. find the value of ( j + k ).

| ( x ) | 13 | 15 | 17 | 19 | 21 | 23 |
| ( h(x) ) | ( j ) | ( -8 ) | ( -3 ) | ( -1 ) | ( -2 ) | ( k ) |

a) ( -22 )
b) ( -20 )
c) ( -19 )
d) ( -6 )

8. the graph of a smooth, continuous function ( f ) has a single inflection point at ( x = 3 ). which of the following statements is true?
a) ( f ) is increasing at ( x = 3 )
b) the graph of ( f ) is concave down at ( x = 3 )
c) ( f ) has an absolute minimum value at ( x = 3 )
d) the rate of change of ( f ) changes from increasing to decreasing or decreasing to increasing at ( x = 3 )

Accepted Answer

### Question 5
# Brief Explanations:
To determine the correct option, recall the definitions of average and instantaneous rates of change. The average rate of change of a function over an interval $[a, b]$ is $\frac{f(b)-f(a)}{b - a}$, which is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. The instantaneous rate of change at a point $x = c$ is the limit of the average rate of change as the interval shrinks to $c$, which is the slope of the tangent line at $c$.

- Option A: Incorrect. Average rate is secant slope, instantaneous is tangent slope. Also, there's no general rule about speed.
- Option B: Incorrect. Average rate is over an interval (secant slope), instantaneous at a point (tangent slope). This option swaps them.
- Option C: Incorrect. Both can be zero. For example, a function with a maximum has instantaneous rate zero there, and a constant function has average rate zero over any interval.
- Option D: Correct. Average rate of change is calculated over an interval (using two points), and instantaneous rate is at a single point (limit as interval length approaches zero).

# Answer: D

### Question 6
# Explanation:
## Step 1: Define the cost function
The catering cost $C(p)$ has a fixed service fee of \$200 and a variable cost of \$24 per plate. Also, Johanna and the groom (2 people) are not guests, so the number of plates $p$ ranges from the number of confirmed guests (72) plus 2 (for her and groom) to the total possible guests (72 + 18) plus 2. So $p$ is in $[72 + 2, 72 + 18 + 2]=[74, 92]$. The cost function is $C(p)=200 + 24p$.

## Step 2: Calculate the minimum cost
Substitute $p = 74$ into $C(p)$:
$C(74)=200 + 24\times74 = 200 + 1776 = 1976$? Wait, no, wait: Wait, 72 guests, plus Johanna and groom (2), so minimum $p = 72 + 2 = 74$? Wait, no, the problem says "Johanna has gotten RSVPs accepting the invitation from 72 guests, but is still waiting to hear back from the other 18. She also has to pay for her own food and the groom’s food, who are not considered guests." So total plates $p$ is (number of guests who RSVP + number of guests who might RSVP) + 2 (her and groom). So minimum $p = 72 + 2 = 74$ (if no more RSVP), maximum $p = 72 + 18 + 2 = 92$ (if all 18 RSVP). Now calculate $C(74)$: $200 + 24\times74 = 200 + 1776 = 1976$? Wait, no, 24*74: 24*70=1680, 24*4=96, so 1680+96=1776; 1776+200=1976. Maximum $p = 92$: $C(92)=200 + 24\times92$. 24*90=2160, 24*2=48, so 2160+48=2208; 2208+200=2408? Wait, no, wait the options: Option A is $1976 \leq C(p) \leq 2408$, Option C is $1776 \leq C(p) \leq 2208$. Wait, I think I made a mistake. Wait, the service fee is \$200, and per plate is \$24. But the "plates" are for guests plus Johanna and groom. Wait, maybe the 72 guests are the ones who RSVP, and the 18 are pending. So the number of plates $p$ is (72 + 0 to 18) + 2 (Johanna and groom). So minimum $p = 72 + 2 = 74$? No, wait, maybe the 72 guests are the ones who accepted, so their plates are 72, plus Johanna and groom: 72 + 2 = 74. The pending 18: if they all accept, total guests are 72 + 18 = 90, plus Johanna and groom: 90 + 2 = 92. So $p$ is 74 to 92. Now $C(p)=200 + 24p$. Wait, but 24*74 = 1776, plus 200 is 1976. 24*92 = 2208, plus 200 is 2408. So the range is $1976 \leq C(p) \leq 2408$, which is Option A. Wait, but let's check again. Wait, maybe the "per plate" is for the guests and the couple? Wait, the problem says "cost of food for one guest" is \$24, and she has to pay for her own and groom's food, who are not guests. So the number of plates is (number of guests) + 2 (her and groom). So guests: minimum 72 (RSVP), maximum 72 + 18 = 90. So plates $p$: minimum 72 + 2 = 74, maximum 90 + 2 = 92. Then $C(p)=200 + 24p$. So $C(74)=200 + 24*74 = 200 + 1776 = 1976$. $C(92)=200 + 24*92 = 200 + 2208 = 2408$. So the range is $1976 \leq C(p) \leq 2408$, which is Option A.

# Answer: A

### Question 7
# Explanation:
## Step 1: Recall properties of quadratic functions
A quadratic function has the form $h(x)=ax^2 + bx + c$, and its graph is a parabola, which is symmetric about its vertex. The axis of symmetry is the vertical line through the vertex, so the function values are symmetric around the vertex's $x$-coordinate.

## Step 2: Find the axis of symmetry
Looking at the table, the $x$-values are 13, 15, 17, 19, 21, 23. Let's check the symmetry. The middle of 17 and 19 is 18, but let's check the differences. Wait, the values at $x = 17$ is $-8$, $x = 19$ is $-1$? Wait no, the table is:

$x$: 13, 15, 17, 19, 21, 23

$h(x)$: $j$, $-8$, $-3$, $-1$, $-2$, $k$

Wait, that can't be a quadratic because the second differences should be constant. Wait, maybe I misread. Wait, quadratic functions have symmetric values around the vertex. Let's list the $x$-values and their distances from the vertex. Let's assume the vertex is at $x = 19$ (since the values around $x = 19$ might be symmetric). Wait, from $x = 17$ to $x = 19$: difference 2, $h(x)$ goes from $-3$ to $-1$ (increase by 2). From $x = 19$ to $x = 21$: difference 2, $h(x)$ goes from $-1$ to $-2$ (decrease by 1). That's not symmetric. Wait, maybe the vertex is at $x = 20$? Wait, the $x$-values are 13,15,17,19,21,23. The midpoint between 17 and 21 is 19? No, 17 and 21: (17+21)/2=19. Midpoint between 15 and 23: (15+23)/2=19. Midpoint between 13 and 25 (but 25 isn't there), but 13 and 23: (13+23)/2=18. Wait, maybe the axis of symmetry is $x = 19$? Wait, no, let's check the second differences. For a quadratic, the second difference (difference of differences) is constant.

First, list the $x$ and $h(x)$:

$x$: 13, 15, 17, 19, 21, 23

$h(x)$: $j$, $-8$, $-3$, $-1$, $-2$, $k$

First differences (Δh) between consecutive $x$ (difference in $x$ is 2 each time, so we can treat it as equally spaced):

Between $x=13$ and $x=15$: $-8 - j$

Between $x=15$ and $x=17$: $-3 - (-8) = 5$

Between $x=17$ and $x=19$: $-1 - (-3) = 2$

Between $x=19$ and $x=21$: $-2 - (-1) = -1$

Between $x=21$ and $x=23$: $k - (-2) = k + 2$

Second differences (Δ²h) are differences of first differences:

Between first and second first differences: $5 - (-8 - j) = 13 + j$

Between second and third first differences: $2 - 5 = -3$

Between third and fourth first differences: $-1 - 2 = -3$

Between fourth and fifth first differences: $(k + 2) - (-1) = k + 3$

For a quadratic, second differences should be constant. So the second differences from $x=15$ to $x=17$ to $x=19$ to $x=21$ should be equal. The second differences between $x=17$-$x=19$ and $x=19$-$x=21$ are both $-3$, so the second difference should be $-3$ throughout.

So, the second difference between $x=13$-$x=15$ and $x=15$-$x=17$ should be $-3$:

$13 + j = -3$ → $j = -16$

The second difference between $x=21$-$x=23$ and $x=19$-$x=21$ should be $-3$:

$k + 3 = -3$ → $k = -6$

Now, find $j + k$: $j = -16$, $k = -6$, so $j + k = -16 + (-6) = -22$? Wait, but let's check again. Wait, maybe the axis of symmetry is at $x = 19$, so the value at $x = 17$ should equal the value at $x = 21$ if symmetric, but $h(17) = -3$ and $h(21) = -2$, which is not equal. Wait, maybe I made a mistake. Wait, let's try another approach. Let's assume the quadratic is $h(x) = ax² + bx + c$. We have points:

At $x=15$: $225a + 15b + c = -8$

At $x=17$: $289a + 17b + c = -3$

At $x=19$: $361a + 19b + c = -1$

Subtract first equation from second: $64a + 2b = 5$ (Equation 1)

Subtract second equation from third: $72a + 2b = 2$ (Equation 2)

Subtract Equation 1 from Equation 2: $8a = -3$ → $a = -\frac{3}{8}$

Plug $a = -\frac{3}{8}$ into Equation 1: $64*(-\frac{3}{8}) + 2b = 5$ → $-24 + 2b = 5$ → $2b = 29$ → $b = \frac{29}{2}$

Now, find $c$ from first equation: $225*(-\frac{3}{8}) + 15*(\frac{29}{2}) + c = -8$

Calculate: $-\frac{675}{8} + \frac{435}{2} + c = -8$ → $-\frac{675}{8} + \frac{1740}{8} + c = -8$ → $\frac{1065}{8} + c = -8$ → $c = -8 - \frac{1065}{8} = -\frac{64}{8} - \frac{1065}{8} = -\frac{1129}{8}$

Now, find $h(13)$ (which is $j$): $x=13$: $169a + 13b + c$

Plug in $a, b, c$:

$169*(-\frac{3}{8}) + 13*(\frac{29}{2}) - \frac{1129}{8}$

= $-\frac{507}{8} + \frac{377}{2} - \frac{1129}{8}$

= $-\frac{507}{8} + \frac{1508}{8} - \frac{1129}{8}$

= $\frac{-507 + 1508 - 1129}{8}$

= $\frac{-128}{8} = -16$ → $j = -16$

Find $h(23)$ (which is $k$): $x=23$: $529a + 23b + c$

= $529*(-\frac{3}{8}) + 23*(\frac{29}{2}) - \frac{1129}{8}$

= $-\frac{1587}{8} + \frac{667}{2} - \frac{1129}{8}$

= $-\frac{1587}{8} + \frac{2668}{8} - \frac{1129}{8}$

= $\frac{-1587 + 2668 - 1129}{8}$

= $\frac{-48}{8} = -6$ → $k = -6$

Thus, $j + k = -16 + (-6) = -22$

# Answer: A

### Question 8
# Brief Explanations:
Recall the definition of an inflection point: a point where the concavity of the function changes (i.e., the second derivative changes sign). The first derivative $f'$ represents the rate of change of $f$, and the second derivative $f''$ represents the rate of change of $f'$ (concavity of $f$).

- Option A: An inflection point doesn't tell us about whether $f$ is increasing or decreasing. It's about concavity, not the sign of $f'$. So A is incorrect.
- Option B: An inflection point is where concavity changes, so we can't say the graph is concave down at $x = 3$ (it could be concave up before and down after, or vice versa). So B is incorrect.
- Option C: An inflection point is not related to absolute minima (which are related to critical points and the first derivative). So C is incorrect.
- Option D: The rate of change of $f$ is $f'$. The inflection point is where $f''$ (the derivative of $f'$) changes sign, meaning $f'$ changes from increasing to decreasing (if $f''$ goes from positive to negative) or decreasing to increasing (if $f''$ goes from negative to positive). So D is correct.

# Answer: D

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QUESTION IMAGE

Question

Explanation:

Question 5

Step 1: Define the cost function

Step 2: Calculate the minimum cost

Step 1: Recall properties of quadratic functions

Step 2: Find the axis of symmetry

Answer:

Question 6