Chapter 5

Incidence and prevalence The incidence \(i(t)\) of an infectious disease at time \(t\) is the rate at which new infections are occurring at that time. The prevalence \(P(t)\) at time \(t\) is the total number of infected individuals at that time. Let's suppose that \(P(0)=0\) . (a) Express the total number of new infections between times \(t=0\) and \(t=a\) as a definite integral. (b) Suppose that all individuals either die or recover from infection, and that \(D\) is the total number that have done so between times \(t=0\) and \(t=a .\) Express \(D\) in terms of \(P(a)\) and your result from part (a). (c) Let \(d(t)\) be the rate at which people are dying or recovering from infection at time \(t .\) What is the relationship between \(D\) and \(d(t) ?\)

Photosynthesis Much of the earth's photosynthesis occurs in the oceans. The rate of primary production (as discussed in Exercise 61 depends on light intensity, measured as the flux of photons (that is, number of photons per unit area per unit time). For monochromatic light, intensity decreases with water depth according to Beer's Law, which states that \(I(x)=e^{-k x}\) , where \(x\) is water depth. A simple model for the relationship between rate of photosynthesis and light intensity is \(P(I)=a I,\) where \(a\) is a constant and \(P\) is measured as a mass of carbon fixed per volume of water, per unit time. (a) What is the rate of photosynthesis as a function of water depth? (b) What is the total rate of photosynthesis of a water column that is one unit in surface area and four units deep? (c) What is the total rate of photosynthesis of a water column that is one unit in surface area and \(x\) units deep? (d) What is the rate of change of the total photosynthesis with respect to the depth \(x\) ?

Photosynthesis The rate of primary production refers to the rate of conversion of inorganic carbon to organic carbon via photosynthesis. It is measured as a mass of carbon fixed per unit biomass, per unit time. A common model for this relationship is $$P(I)=\frac{a I}{\sqrt{1+b I^{2}}}$$ where \(P\) is the rate of primary production as a function of light intensity \(I\) . Suppose the light intensity changes with time according to the equation \(I(t)=k t,\) where \(k\) is a constant (a) What is the rate of primary production as a function of time? (b) What is the total amount of primary production over the first five units of time?

(a) Show that \(1 \leqslant \sqrt{1+x^{3}} \leqslant 1+x^{3}\) for \(x \geqslant 0\) . (b) Show that 1\(\leqslant \int_{0}^{1} \sqrt{1+x^{3}} d x \leqslant 1.25\)

Growing degree days The rate of development of many plant species depends on the temperature of the environment in such a way that maturity is reached only after a certain number of "degree-days." Suppose that temperature \(T\) as a function of time \(t\) is given by $$T(t)=15\left(1+\sin \frac{2 \pi t}{60}\right) \quad 0 \leqslant t \leqslant 60$$ where \(t\) is measured in days. If maturity is reached on day\(t=20,\) what is the number of degree-days required? \(\left[\) Hint: What are the units for \(\int_{0}^{20} T(t) d t ?\)]$

(a) Show that \(\cos \left(x^{2}\right) \geqslant \cos x\) for 0\(\leqslant x \leqslant 1\) (b) Deduce that \(\int_{0}^{\pi / 6} \cos \left(x^{2}\right) d x \geqslant \frac{1}{2}\)

If \(f\) is continuous and \(\int_{0}^{9} f(x) d x=4,\) find \(\int_{0}^{3} x f\left(x^{2}\right) d x\)

If \(f\) is continuous and \(\int_{0}^{4} f(x) d x=10,\) find \(\int_{0}^{2} f(2 x) d x\)

If \(f\) is continuous on \(\mathbb{R},\) prove that $$\int_{a}^{b} f(x+c) d x=\int_{a+c}^{b+c} f(x) d x$$ For the case where \(f(x) \geqslant 0,\) draw a diagram to interpret this equation geometrically as an equality of areas.

\(67-68\) Sketch the area represented by \(g(x) .\) Then find \(g^{\prime}(x)\) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. \(g(x)=\int_{0}^{x}\left(1+t^{2}\right) d t\)

If \(a\) and \(b\) are positive numbers, show that $$\int_{0}^{1} x^{a}(1-x)^{b} d x=\int_{0}^{1} x^{b}(1-x)^{a} d x$$

Sketch the area represented by \(g(x) .\) Then find \(g^{\prime}(x)\) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. \(g(x)=\int_{0}^{x}(1+\sqrt{t}) d t\)

\(69-78\) Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(g(x)=\int_{1}^{x} \frac{1}{t^{3}+1} d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(g(x)=\int_{3}^{x} e^{t^{2}-t} d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(g(y)=\int_{2}^{y} t^{2} \sin t d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(g(r)=\int_{0}^{r} \sqrt{x^{2}+4} d x\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.\(G(x)=\int_{x}^{1} \cos \sqrt{t} d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.\(h(x)=\int_{2}^{1 / x} \arctan t d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(h(x)=\int_{0}^{x^{2}} \sqrt{1+r^{3}} d r\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(y=\int_{0}^{\tan x} \sqrt{t+\sqrt{t}} d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(y=\int_{e^{x}}^{0} \sin ^{3} t d t\)

If \(f(1)=12, f^{\prime}\) is continuous, and \(\int_{1}^{4} f^{\prime}(x) d x=17,\) what is the value of \(f(4) ?\)

Suppose \(h\) is a function such that \(h(1)=-2, h^{\prime}(1)=2\) \(h^{\prime \prime}(1)=3, h(2)=6, h^{\prime}(2)=5, h^{\prime \prime}(2)=13,\) and \(h^{\prime \prime}\) is continuous everywhere. Evaluate \(\int_{1}^{2} h^{\prime \prime}(u) d u\)

\(83-84\) Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on \([0,1]\) . \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{i^{3}}{n^{4}}\)

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on \([0,1]\) . \(\lim _{n \rightarrow \infty} \frac{1}{n}\left(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\sqrt{\frac{3}{n}}+\cdots+\sqrt{\frac{n}{n}}\right)\)

Find a function \(f\) and a number \(a\) such that $$\quad 6+\int_{a}^{x} \frac{f(t)}{t^{2}} d t=2 \sqrt{x} \quad\( for all \)x>0$$