Chapter 6

Consider a population whose size at time \(t\) is \(N(t)\) and whose growth obeys the initial-value problem $$ \frac{d N}{d t}=e^{-t} $$ with \(N(0)=100\). (a) Find \(N(t)\) by solving the initial-value problem. (b) Compute the cumulative change in population size between \(t=0\) and \(t=5\). (c) Express the cumulative change in population size between time 0 and time \(t\) as an integral. Give a geometric interpretation of this quantity.

In Problems \(1-14\), find \(\frac{d y}{d x}\) \(y=\int_{0}^{x} 2 t^{2} d t\)

Approximate the area under the parabola \(y=x^{2}\) from 0 to 1 , using four equal subintervals.

Suppose that a change in biomass \(B(t)\) at time \(t\) during the interval \([0,12]\) follows the equation $$ \frac{d B}{d t}(t)=\cos \left(\frac{\pi}{6} t\right) $$ for \(0 \leq t \leq 12\) (a) Graph \(\frac{d B}{d r}\) as a function of \(t\). (b) Suppose that \(B(0)=B_{0} .\) Express the cumulative change in biomass during the interval \([0, t]\) as an integral. Give a geometric interpretation. What is the value of the biomass at the end of the interval \([0,12]\) compared with the value at time 0 ? How are these two quantities related to the cumulative change in the biomass during the interval \([0,12] ?\)

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x}\left(4-\frac{t^{4}}{2}\right) d t\)

Approximate the area under the parabola \(y=x^{2}\) from 0 to 1 . using five equal subintervals.

A particle moves along the \(x\) -axis with velocity $$ v(t)=-(t-2)^{2}+1 $$ for \(0 \leq t \leq 5\). Assume that the particle is at the origin at time 0 . (a) Graph \(v(t)\) as a function of \(t\). (b) Use the graph of \(v(t)\) to determine when the particle moves to the left and when it moves to the right. (c) Find the location \(s(t)\) of the particle at time \(t\) for \(0 \leq t \leq 5\). Give a geometric interpretation of \(s(t)\) in terms of the graph of \(v(t) .\) (d) Graph \(s(t)\) and find the leftmost and rightmost positions of the particle.

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x}(4 t-3) d t\)

Approximate the area under the curve \(y=x^{3}\) from 0 to 1 , using six equal subintervals.

Recall that the acceleration \(a(t)\) of a particle moving along a straight line is the instantaneous rate of change of the velocity \(v(t) ;\) that is, $$ a(t)=\frac{d}{d t} v(t) $$ Assume that \(a(t)=32 \mathrm{ft} / \mathrm{s}^{2}\). Express the cumulative change in velocity during the interval \([0, t]\) as a definite integral, and compute the integral.

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x}\left(3+t^{4}\right) d t\)

Approximate the area under the parabola \(y=1-x^{2}\) from 0 to 1, using five equal subintervals.

If \(\frac{d l}{d}\) represents the growth rate of an organism at time \(t\) (measured in months), explain what $$ \int_{2}^{7} \frac{d l}{d t} d t $$ represents.

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x} \sqrt{1+2 t} d t, x>\frac{-1}{2}\)

Approximate the area under the curve \(y=x^{3}-x\) from 0 to 1 using five equal subintervals.

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x} \sqrt{1+t^{2}} d t\)

Approximate the area under the curve \(y=x^{2}-x\) from 0 to 1 using six equal subintervals. In Problems 7 and 8, you will use Riemann sums to prove that \(\int_{0}^{a} x^{2} d x=\frac{1}{3} a^{3} .\)

If \(\frac{d B}{d}\) represents the rate of change of biomass of a plant at time \(t\), explain what $$ \int_{1}^{6} \frac{d B}{d t} d t $$ means.

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x} \sqrt{\sin 2 t} d t, 0

. (a) Let the points \(x_{0}=0

Let \(N(t)\) denote the size of a population at time \(t\), and assume that $$ \frac{d N}{d t}=f(t) $$ Express the cumulative change of the population size in the interval \([0,3]\) as an integral.

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x} \cos (t+1) d t\)

Find \(\frac{d y}{d x}\) \(y=\int_{3}^{x} t e^{4 t} d t\)

Approximate $$ \int_{-1}^{1}\left(1-x^{2}\right) d x $$ using five equal subintervals.

Fish Growth The rate of growth of a fish is sometimes modeled by the equation $$ d L / d t=L_{0} e^{-k t} $$ where \(L\) is the length of the fish, and \(k\) and \(L_{0}\) are positive constants. (a) Interpret in words the quantity \(\int_{0}^{3} \frac{d L}{d f} d t\). (b) Calculate the integral from part (a); your answer will include the constants \(k\) and \(L_{0}\).

Find \(\frac{d y}{d x}\) \(y=\int_{1}^{x} t e^{-t^{2}} d t\)

Approximate $$ \int_{-1}^{1}\left(1+x^{2}\right) d x $$ using five equal subintervals.

Let \(f(x)=x^{2}-2 .\) Compute the average value of \(f(x)\) over the interval \([0,2]\).

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x} \frac{1}{t+1} d t, x>-1\)

Approximate $$ \int_{-1}^{1}\left(2+x^{2}\right) d x $$ using five equal subintervals.

Let \(g(t)=\sin (\pi t)\). Compute the average value of \(g(t)\) over the interval \([-1,1]\).

Find \(\frac{d y}{d x}\) \(y=\int_{-1}^{x} \frac{2}{t^{2}+t} d t\)

Approximate $$ \int_{-2}^{2}\left(2+x^{2}\right) d x $$ using six equal subintervals.

Suppose that the temperature \(T\) (measured in degrees Fahrenheit) in a growing chamber varies over a 24 -hour period according to $$ T(t)=68+\sin \left(\frac{\pi}{12} t\right) $$ for \(0 \leq t \leq 24\). (a) Graph the temperature \(T\) as a function of time \(t\). (b) Find the average temperature and explain your answer graphically.

Find \(\frac{d y}{d x}\) $$ y=\int_{\pi / 2}^{x} \sin \left(t^{2}+1\right) d t $$

Approximate $$ \int_{-1}^{2} e^{-x} d x $$ using three equal subintervals.

Suppose that the concentration (measured in \(\mathrm{gm}^{-3}\) ) of nitrogen in the soil along a transect in moist tundra yields data points that follow a straight line with equation $$ y=673.8-34.7 x $$ for \(0 \leq x \leq 10\), where \(x\) is the distance to the beginning of the transect. What is the average concentration of nitrogen in the soil along this transect?

Find \(\frac{d y}{d x}\) $$ y=\int_{x / 4}^{x} \cos ^{2}(t-3) d t $$

Approximate $$ \int_{0}^{\pi} \sin x d x $$ using four equal subintervals.

In Problems 15-38, use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x}\left(1+t^{2}\right) d t $$

The average daily temperature (measured in Fahrenheit) in New York city can be approximated by the following function of the time of year \(t .(t\) measures the fraction of the year that has elapsed since January 1.) $$ T(t)=57.5-22.5 \cos (2 \pi t) $$ (a) Sketch the function \(T(t)\) against \(t\). (b) What is the average daily temperature high, averaged over the course of one year? (c) Explain how you could get your answer in part (b) without doing any integrations. (d) What is the average summer temperature? You may assume that summer corresponds to the interval \(0.47 \leq t \leq 0.73 .\) You will need to use a calculator to evaluate your answer.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{2 x-1}\left(t^{3}-1\right) d t $$

Assume that \(a<0

The typical daily temperature high, measured in \({ }^{\circ} C\) (degrees Celsius), in Los Angeles varies over the course of a year according to the formula \(T(t)=21.7+3.1 \cos (2 \pi(t-0.75))\) (where \(t\) measures the fraction of the year that has elapsed since January 1). (a) Sketch the graph of the function \(T(t)\). (b) What is the average daily temperature high, averaged over the course of one year? (c) Explain how you could get your answer in part (b) without doing any integrations. (d) What is the average winter temperature? You may assume that winter corresponds to the interval \(0 \leq t \leq 0.22\) and \(0.98 \leq\) \(t \leq 1\). You will need to use a calculator to evaluate your answer.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{1-4 x}\left(2 t^{2}+1\right) d t $$

Suppose that you drive from St. Paul to Duluth and you average \(50 \mathrm{mph}\). Explain why there must be a time during your trip at which your speed is exactly \(50 \mathrm{mph}\).

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x+2} t(1+t) d t $$

Let \(f(x)=2 x, 0 \leq x \leq 2 .\) Use a geometric argument to find the average value of \(f\) over the interval \([0,2]\), and find \(x\) such that \(f(x)\) is equal to this average value.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{4}^{x^{2}+1} \sqrt{t} d t $$

A particle moves along the \(x\) -axis with velocity $$ v(t)=-(t-3)^{2}+5 $$ for \(0 \leq t \leq 6\). (a) Graph \(v(t)\) as a function of \(t\) for \(0 \leq t \leq 6\). (b) Find the average velocity of this particle during the interval \([0,6] .\) (c) Find a time \(t^{*} \in[0,6]\) such that the velocity at time \(t^{*}\) is equal to the average velocity during the interval \([0,6] .\) Is it clear that such a point exists? Is there more than one such point in this case? Use your graph in (a) to explain how you would find \(t^{*}\) graphically.