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Section 1
Antiderivatives Graphically and Numerically
Fill in the blanks in the following statements, assuming that $F(x)$ is an antiderivative of $f(x)$ :(a) If $f(x)$ is positive over an interval, then $F(x)$ is ______ over the interval.(b) If $f(x)$ is increasing over an interval, then $F(x)$ is ______ over the interval.
Use Figure 6.10 and the fact that $P=0$ when $t=0$ to find values of $P$ when $t=1,2,3,4$ and 5
Use Figure 6.11 and the fact that $P=2$ when $t=0$ to find values of $P$ when $t=1,2,3,4$ and $5 .$
Let $G^{\prime}(t)=g(t)$ and $G(0)=4 .$ Use Figure 6.12 to find the values of $G(t)$ at $t=5,10,20,25$.
Sketch two functions $F$ such that $F^{\prime}=f$. In one case let $F(0)=0$ and in the other, let $F(0)=1$.
Let $F(x)$ be an antiderivative of $f(x),$ with $F(0)=50$ and $\int_{0}^{5} f(x) d x=12 .$ What is $F(5) ?$
Let $F(x)$ be an antiderivative of $f(x),$ with $F(1)=20$ and $\int_{1}^{4} f(x) d x=-7 .$ What is $F(4) ?$
Let $F(x)$ be an antiderivative of $f(x)$.(a) If $\int_{2}^{5} f(x) d x=4$ and $F(5)=10,$ find $F(2)$(b) If $\int_{0}^{100} f(x) d x=0,$ what is the relationship between $F(100)$ and $F(0) ?$
(a) Estimate $\int_{0}^{4} f(x) d x$ for $f(x)$ in Figure 6.13(b) Let $F(x)$ be an antiderivative of $f(x)$. Is $F(x)$ increasing or decreasing on the interval $0 \leq x \leq 4 ?$(c) If $F(0)=20,$ what is $F(4) ?$
(a) Estimate $\int_{0}^{4} f(x) d x$ for $f(x)$ in Figure 6.14(b) Let $F(x)$ be an antiderivative of $f(x)$. If $F(0)=$ $100,$ what is $F(4) ?$
If $F(0)=5$ and $F(x)$ is an antiderivative of $f(x)=$ $3 e^{-x^{2}},$ use a calculator to find $F(2)$.
If $G(1)=50$ and $G(x)$ is an antiderivative of $g(x)=$ $\ln x,$ use a calculator to find $G(4)$.
Estimate $f(x)$ for $x=2,4,6,$ using the given values of $f^{\prime}(x)$ and the fact that $f(0)=100$.$$\begin{array}{l|c|c|c|c}\hline x & 0 & 2 & 4 & 6 \\\hline f^{\prime}(x) & 10 & 18 & 23 & 25 \\\hline\end{array}$$
Estimate $f(x)$ for $x=2,4,6,$ using the given values of $f^{\prime}(x)$ and the fact that $f(0)=50$.$$\begin{array}{l|r|r|r|r}\hline x & 0 & 2 & 4 & 6 \\\hline f^{\prime}(x) & 17 & 15 & 10 & 2 \\\hline\end{array}$$
Using Figure $6.15,$ sketch a graph of an antiderivative $G(t)$ of $g(t)$ satisfying $G(0)=5 .$ Label each critical point of $G(t)$ with its coordinates.
Use Figure 6.16 and the fact that $F(2)=3$ to sketch the graph of $F(x)$. Label the values of at least four points.
Figure 6.17 shows the rate of change of the concentration of adrenaline, in micrograms per milliliter per minute, in a person's body. Sketch a graph of the concentration of adrenaline, in micrograms per milliliter, in the body as a function of time, in minutes.
The graph in Figure 6.18 records the spillage rate at a toxic waste treatment plant over the 50 minutes it took to plug the leak.(a) Complete the table for the total quantity spilled in liters in time $t$ minutes since the spill started.$$\begin{array}{l|llllll}\hline \text { Time } t(\min ) & 0 & 10 & 20 & 30 & 40 & 50 \\\hline \text { Quantity (liters) } & 0 & & & & & \\\hline\end{array}$$
Two functions, $f(x)$ and $g(x)$, are shown in Figure 6.19 . Let $F$ and $G$ be antiderivatives of $f$ and $g,$ respectively. On the same axes, sketch graphs of the antiderivatives $F(x)$ and $G(x)$ satisfying $F(0)=0$ and $G(0)=0 .$ Compare $F$ and $G,$ including a discussion of zeros and $x$ and $y$ -coordinates of critical points.
Let $F(x)$ be an antiderivative of $f(x)=1-x^{2}$(a) On what intervals is $F(x)$ increasing?(b) On what intervals is the graph of $F(x)$ concave up?
Sketch two functions $F$ with $F^{\prime}(x)=$ $f(x) .$ In one, let $F(0)=0 ;$ in the other, let $F(0)=1 .$ Mark $x_{1}, x_{2},$ and $x_{3}$ on the $x$ -axis of your graph. Identify local maxima, minima, and inflection points of $F(x)$.
A particle moves back and forth along the $x$ -axis. Figure 6.20 approximates the velocity of the particle as a function of time. Positive velocities represent movement to the right and negative velocities represent movement to the left. The particle starts at the point $x=5 .$ Graph the distance of the particle from the origin, with distance measured in kilometers and time in hours.
Assume $f^{\prime}$ is given by the graph in Figure $6.21 .$ Suppose $f$ is continuous and that $f(0)=0$.(a) Find $f(3)$ and $f(7)$(b) Find all $x$ with $f(x)=0$.(c) Sketch a graph of $f$ over the interval $0 \leq x \leq 7$.
Urologists are physicians who specialize in the health of the bladder. In a common diagnostic test, urologists monitor the emptying of the bladder using a device that produces two graphs. In one of the graphs the flow rate (in milliliters per second) is measured as a function of time (in seconds). In the other graph, the volume emptied from the bladder is measured (in milliliters) as a function of time (in seconds). See Figure 6.22 .(a) Which graph is the flow rate and which is the volume?(b) Which one of these graphs is an antiderivative of the other?
The Quabbin Reservoir in the western part of Massachusetts provides most of Boston's water. The graph in Figure 6.23 represents the flow of water in and out of the Quabbin Reservoir throughout 2016 .(a) Sketch a graph of the quantity of water in the reservoir, as a function of time.(b) When, in the course of $2016,$ was the quantity of water in the reservoir largest? Smallest? Mark and label these points on the graph you drew in part (a).(c) When was the quantity of water increasing most rapidly? Decreasing most rapidly? Mark and label these times on both graphs.(d) By July 2017 the quantity of water in the reservoir was about the same as in January 2016 . Draw plausible graphs for the flow into and the flow out of the reservoir for the first half of 2017 .
The birth rate, $B$, in births per hour, of a bacteria population is given in Figure $6.24 .$ The curve marked $D$ gives the death rate, in deaths per hour, of the same population.(a) Explain what the shape of each of these graphs tells you about the population.(b) Use the graphs to find the time at which the net rate of increase of the population is at a maximum.(c) At time $t=0$ the population has size $N$. Sketch the graph of the total number born by time $t$. Also sketch the graph of the number alive at time $t .$ Estimate the time at which the population is a maxi-mum.
Explain what is wrong with the statement.Let $F(x)$ be an antiderivative of $f(x)$. If $f(x)$ is everywhere increasing, then $F(x) \geq 0$.
Explain what is wrong with the statement.If $F(x)$ and $G(x)$ are both antiderivatives of $f(x),$ then $H(x)=F(x)+G(x)$ must also be an antiderivative of $f(x)$.
Give an example of:A graph of a function $f(x)$ such that $\int_{0}^{2} f(x) d x=0$.
Give an example of:A graph of a function $f(x)$ whose antiderivative is increasing everywhere.
True or false? Give an explanation for your answer.A function $f(x)$ has at most one derivative.
True or false? Give an explanation for your answer.If $f(t)$ is a linear function with positive slope, then an antiderivative, $F,$ is a linear function.