Chapter 6

Write each sum in sigma notation. $$ \frac{3}{5}+\frac{4}{6}+\frac{5}{7}+\frac{6}{8}+\frac{7}{9} $$

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

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.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2}^{x^{2}-2} \sqrt{3+u} d u, x>0 $$

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.

Write each sum in sigma notation. $$ \frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots+\frac{1}{2^{n}} $$

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

If \(\frac{d l}{d t}\) 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.

Write each sum in sigma notation. $$ 1+q+q^{2}+q^{3}+q^{4}+\cdots+q^{n-1} $$

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

If \(\frac{d w}{d x}\) represents the rate of change of the weight of an organism of age \(x\), explain what $$ \int_{3}^{5} \frac{d w}{d x} d x $$ means.

Write each sum in sigma notation. $$ 1-a+a^{2}-a^{3}+a^{4}-a^{5}+\cdots+(-1)^{n} a^{n} $$

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

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

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=1}^{15}(2 k+3) $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2}^{\ln x} e^{-t} d t, x>0 $$

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.

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=1}^{5}\left(4-k^{2}\right) $$

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

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

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=0}^{6} k(k+1) $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x}^{5}\left(1+e^{t}\right) d t $$

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

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=1}^{n} 4 k $$

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

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.

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=1}^{n} 4(k-1)^{2} $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2 x^{2}}^{6}(1+\tan t) d t $$

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?

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=1}^{n}(k+2)(k-2) $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x}^{5} \frac{1}{u^{2}} d u, x>0 $$

Let \(f(x)=\tan x\). Give a geometric argument to explain why the average value of \(f(x)\) over \([-1,1]\) is equal to 0 .

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=1}^{10}(-1)^{k} $$

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

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 the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$ \sum_{k=0}^{10}(-1)^{k} $$

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

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 \(\frac{d y}{d x}\). $$ y=\int_{2+x^{2}}^{2} \cot t d t $$

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.

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

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

Find the volume of a right circular cone with base radius \(r\) and height \(h\).

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

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{-x}^{x} \tan u d u, 0

Find the volume of a pyramid with square base of side length \(a\) and height \(\bar{h}\).

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

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

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

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x^{3}}^{x^{4}} \ln \left(1+t^{2}\right) d t, x>0 $$