Solved: ?Let \(g(x)=\int_{0}^{x} f(t) d t\), where \(f\) is the function whose graph is

Explain exactly what is meant by the statement that "differentiation and integration are inverse processes."

Let \(g(x)=\int_{0}^{x} f(t) d t\), where \(f\) is the function whose graph is shown. (a) Evaluate \(g(x)\) for \(x=0,1,2,3,4,5\), and \(6\) . (b) Estimate \(g(7)\). (c) Where does \(g\) have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of \(g\). Equation Transcription: Text Transcription: g(x)=integral over 0 ^x f(t)dt f g(x) x=0,1,2,3,4,5,6 g(7) g g

Let \(g(x)=\int_{0}^{x} f(t) d t\), where f is the function whose graph is shown. (a) Evaluate g(0). g(1), g(2), g(3) and g(6) (b) On what interval is g increasing? (c) Where does g have a maximum value? (d) Sketch a rough graph of g.

Let \(g(x)=\int_{0}^{x} f(t) d t\), where \(f\) is the function whose graph is shown. (a) Use Part 1 of the Fundamental Theorem of Calculus to graph \(g^{\prime} .\) (b) Find \(g(3), g^{\prime}(3)\), and \(g^{\prime \prime}(3)\). (c) Does \(g\) have a local maximum, a local minimum, or neither at \(x=6\)? (d) Does \(g\) have a local maximum, a local minimum, or neither at \(x=9\) ? Equation Transcription: Text Transcription: g(x)=integral over 0 ^x (t)dt f g' g(3),g'(3),g"(3) g x=6 g x=9

The graph of a function \(f\) is shown. Let \(g\) be the function that represents the area under the graph of \(f\) between \(0\) and \(x\). (a) Use geometry to find a formula for \(g(x)\). (b) Verify that \(g\) is an antiderivative of \(f\) and explain how this confirms Part 1 of the Fundamental Theorem of Calculus for the function \(f\). Equation Transcription: Text Transcription: f g f 0 x g(x) g f f

The graph of a function \(f\) is shown. Let \(g\) be the function that represents the area under the graph of \(f\) between \(0\) and \(x\). (a) Use geometry to find a formula for \(g(x)\). (b) Verify that \(g\) is an antiderivative of \(f\) and explain how this confirms Part 1 of the Fundamental Theorem of Calculus for the function \(f\). Equation Transcription: Text Transcription: f g f 0 x g(x) g f f

Sketch the area represented by \(g(x)\). Then find \(g’(x)\) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. \(g(x)=\int_{1}^{x} t^{2} \ d t\) Equation Transcription: g(x) g’(x) g(x) = ? t2 dt Text Transcription: g(x) g prime(x) g(x) = integral_1 ^x t^2 dt

Sketch the area represented by . Then find in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. \(g(x)=\int_{0}^{x}(2+\sin t) d t\) Equation Transcription: ? Text Transcription: g(x) g’(x) g(x)=Integral 0^x (2+sint) dt

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

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(g(x)=\int_{1}^{x} \ln \left(1+t^{2}\right) d t\) Equation Transcription: ? Text Transcription: g(x)=Integral from 1^x ln (1+t^2) dt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(g(w)=\int_{0}^{w} \sin \left(1+t^{3}\right) d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(h(u)=\int_{0}^{u} \frac{\sqrt{t}}{t+1} d t\) Equation Transcription: Text Transcription: h(u)=integral over 0 ^u sqrt t/t+1dt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(F(x)=\int_{x}^{0} \sqrt{1+\sec t} d t\) Hint: \(\int_{x}^{0} \sqrt{1+\sec t} d t=-\int_{0}^{x} \sqrt{1+\sec t} d t\) Equation Transcription: ? ? -? Text Transcription: F(x)=Integral from x^0 \sqrt 1+sec t dt Integral from x^0 \sqrt 1+sec t dt=-Integral from 0^x \sqrt 1+sec t dt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(A(w)=\int_{w}^{-1} e^{t+t^{2}} \ d t\) Equation Transcription: A(w) = ? dt Text Transcription: A(w) = integral_w ^-1 e^t+t^2 dt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(h(x)=\int_{1}^{e^{x}} \ln t d t\) Equation Transcription: ? Text Transcription: h(x)=Integral from 1^e^e lntdt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(h(x)=\int_1^{\sqrt{x}} \frac{z^2}{z^4+1} d z\)

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

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(y=\int_{0}^{\operatorname{tan\ x}} \ e^{-t^{2}} \ d t\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \(y=\int_{\sqrt{x}}^{\pi / 4} \theta \tan \theta d \theta\)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral and interpret the result as an area or a difference of areas. Illustrate with a sketch.

Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral and interpret the result as an area or a difference of areas. Illustrate with a sketch. \(\int_{0}^{4}\left(x^{2}-4 x\right) \ d x\) Equation Transcription: ? (x2 - 4x) dx Text Transcription: integral_0 ^4 (x^2 - 4x) dx

Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral and interpret the result as an area or a difference of areas. Illustrate with a sketch. \(\int_{\pi / 2}^{2 \pi}(2 \sin x) d x\) Equation Transcription: dx Text transcription: integral _pi/2 ^2pi (2 sin x) dx

Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral and interpret the result as an area or a difference of areas. Illustrate with a sketch. \(\int_{-1}^{2}\left(e^{x}+2\right) \ d x\) Equation Transcription: ? (ex + 2) dx Text Transcription: integral_-1 ^2 (e^x + 2) dx

Evaluate the integral. \(\int_{1}^{3}\left(x^{2}+2 x-4\right) \ d x\) Equation Transcription: ? (x2 + 2x - 4) dx Text Transcription: integral_1 ^3 (x^2 + 2x - 4) dx

Evaluate the integral.

Evaluate the integral. \(\int_{0}^{2}\left(\frac{4}{5} t^{3}-\frac{3}{4} t^{2}+\frac{2}{5} t\right) \ d t\)

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral. \(\int_{0}^{4}\left(t^{2}+t^{3 / 2}\right) \ d t\) Equation Transcription: ? (t2 + t3/2) dt Text Transcription: integral_0 ^4 (t^2 + t^3/2) dt

Evaluate the integral. \(\int_{1}^{3}\left(\frac{1}{z^{2}}+\frac{1}{z^{3}}\right) \ d z\) Equation Transcription: ? () dz Text Transcription: integral_1 ^3 (1/z^2 + 1/z^3) dz

Evaluate the integral. \(\int_{\pi / 2}^{0} \cos \theta d \theta\)

Evaluate the integral. \(\int_{-5}^5 e d x\)

Evaluate the integral. \(\int_{0}^{1}(u+2)(u-3) \ d u\) Equation Transcription: ? (u + 2)(u - 3) du Text Transcription: integral_ 0 ^1 (u+2)(u-3) du

Evaluate the integral.

Evaluate the integral. \(\int_{1}^{4} \frac{2+x^{2}}{\sqrt{x}} \ d x\) Equation Transcription: ? dx Text Transcription: integral_1 ^4 2+x^2/sqrt 2 dx

Evaluate the integral.

Evaluate the integral. \(\int_{1}^{3}\left(2 x+\frac{1}{x}\right) d x\) Text Transcription: int_{1}^3 (2x +1 / x) dx

Evaluate the integral. \(\int_5^5\sqrt{t^2+\sin t}\ dt\) Equation Transcription: dt Test Transcription: Integral _5 ^5 sqrt t^2 +sin t dt

Evaluate the integral. \(\int_{0}^{\pi / 3} \sec \theta \tan \theta d \theta\) Equation Transcription: Text Transcription: Integral _0 ^pi/6 sec theta tan theta d theta

Evaluate the integral. \(\int_{1}^{3} \ \frac{y^{3}-2 y^{2}-y}{y^{2}} \ d y\) Equation Transcription: ? dy Text Transcription: integral_1 ^3 y^3 - 2y^2 - y/y^2 dy

Evaluate the integral.

Evaluate the integral. \(\int_{0}^{3}\left(2 \sin x-e^{x}\right) d x\) Equation Transcription: dx Text Transcription: Integral _0 ^3 (2 sin x-e^x) dx

Evaluate the integral. \(\int_{1}^{2} \frac{v^{3}+3 v^{6}}{v^{4}} d v\)

Evaluate the integral. \(\int_{1}^{18} \sqrt{\frac{3}{z}} d z\)

Evaluate the integral. \(\int_{0}^{1}\left(x^{e}+e^{x}\right) d x\)

Evaluate the integral. \(\int_{0}^{1} \cosh t d t\) Equation Transcription: dt Text Transcription: Integral _0 ^1 cosh t dt

Evaluate the integral. \(\int_{1 / \sqrt{3}}^{\sqrt{3}} \frac{8}{1+x^{2}} \ d x\) Equation Transcription: ? dx Text Transcription: integral_1/sqrt 3 ^sqrt 3 8/1+x^2 dx

Evaluate the integral. \(\int_{1}^{3} \ \frac{(3 x+1)^{2}}{x^{3}} \ d x\) Equation Transcription: ? dx Text Transcription: integral_1 ^3 (3x + 1)^2/x^3 dx

Evaluate the integral. \(\int_{0}^{4} 2^{s} d s\)

Evaluate the integral. \(\int_{1 / 2}^{1 / \sqrt{2}} \frac{4}{\sqrt{1-x^{2}}} \ d x\) Equation Transcription: ? dx Text Transcription: integral_1/2 ^1/sqrt2 4/sqrt 1-x^2 dx

Evaluate the integral. \(\int_{0}^{\pi} f(x) d x\) where \(f(x)= \begin{cases}\sin x & \text { if } 0 \leqslant x<\pi / 2 \\ \cos x & \text { if } \pi / 2 \leqslant x \leqslant \pi\end{cases}\)

Evaluate the integral.

Sketch the region enclosed by the given curves and calculate its area.

Sketch the region enclosed by the given curves and calculate its area.

Sketch the region enclosed by the given curves and calculate its area.

Sketch the region enclosed by the given curves and calculate its area.

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \(y=\sqrt[3]{x}, 0 \leqslant x \leqslant 27\)

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \(y=\sin x, 0 \leqslant x \leqslant \pi\) Equation Transcription: ?? Text Transcription: y=sinx, 0 leqslant x leqslant pi

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \(y=\sec ^{2} x, 0 \leqslant x \leqslant \pi / 3\) Equation Transcription: ?? Text Transcription: y=sec^2x, 0 leqslant x leqslant pi/3

What is wrong with the equation?

What is wrong with the equation? \(\left.\int_{-1}^{2} \ \frac{4}{x^{3}} \ d x=-\frac{2}{x^{2}}\right]_{-1}^{2}=\frac{3}{2}\) Equation Transcription: ? dx = ] = Text Transcription: integral_-1 ^2 4/x^3 dx = -2/x^2 ]_-1 ^2 = 3/2

What is wrong with the equation? \(\left.\int_{\pi / 3}^{\pi} \sec \theta \tan \theta d \theta=\sec \theta\right]_{\pi / 3}^{\pi}=-3\) Equation Transcription: Text Transcription: Integral _pi/3 ^pi sec theta tan theta d theta=sec theta] _pi/3 ^pi =-3

What is wrong with the equation? \(\left.\int_{0}^{\pi} \sec ^{2} x d x=\tan x\right]_{0}^{\pi}=0\) Equation: Transcription: Text Transcription: Integral _0 ^pi sec^2 x dx=tanx] _0 ^pi =0

Find the derivative of the function. \(g(x)=\int_{2 x}^{3 x} \frac{\mu^{2}-1}{u^{2}+1} d u\) \(\left[\text { Hint }: \int_{2 x}^{3 x} f(u) d u=\int_{2 x}^{0} f(u) d u+\int_{0}^{3 x} f(u) d u\right]\) Equation Transcription: Text Transcription: g(x)=integral _2x ^3x u^2-1/u^2+1 du integral_2x ^3x f(u)du=integral _2x ^0 f(u)du+integral _0 ^3x f(u)du

Find the derivative of the function. \(g(x)=\int_{1-2 x}^{1+2 x} t \sin t d t\) Equation Transcription: Text Transcription: g(x)=integral _1-2x ^1+2x t sin t dt

Find the derivative of the function. \(F(x)=\int_{x}^{x^{2}} e^{t^{2}} \ d t\)

Find the derivative of the function. \(F(x)=\int_{\sqrt{x}}^{2 x} \arctan t d t\) Equation Transcription: Text Transcription: F(x)=integral _sqrt x ^2x arctan t dt

Find the derivative of the function. \(y=\int_{\cos x}^{\sin x} \ln (1+2 v) d v\) Equation Transcription: Text Transcription: y=integral _cos x ^sin x ln(1+2v)dv

If \(f(x)=\int_{0}^{x}\left(1-t^{2}\right) \ e^{t^{2}} \ d t\), on what interval is \(f\) increasing? Equation Transcription: f(x) = ? (1 - t2) dt f Text Transcription: f(x) = integral_0 ^x (1-t^2) e^t^2 dt f

On what interval is the curve \(y=\int_{0}^{x} \frac{t^{2}}{t^{2}+t+2} d t\) concave downward?

Let \(F(x)=\int_{1}^{x} f(t) d t\), where \(f\) is the function whose graph is shown. Where is \(F\) concave downward? ________________ Equation Transcription: Text Transcription: F(x)=\int_{1}^{x} f(t) dt f F

Let \(F(x)=\int_{2}^{x} e^{t^{2}} \ dt\). Find an equation of the tangent line to the curve y = F(x) at the point with x-coordinate 2 .

If \(f(x)=\int_{0}^{\sin x} \sqrt{1+t^{2}} d t\) and \(g(y)=\int_{3}^{y} f(x) d x\), find \(g^{\prime \prime}(\pi / 6)\). Equation Transcription: ’’ Text Transcription: f(x)=integral _0 ^sin x sqrt 1+t^2 dt g(y)=integral _3 ^y f(x) dx g prime prime (pi/6)

Use l'Hospital's Rule to evaluate the limit. \(\lim \limits _{x \rightarrow 0} \ \frac{1}{x^{2}} \ \int_{0}^{x} \ \frac{2 t}{\sqrt{t^{3}+1}} \ d t\) Equation Transcription: ? dt Text Transcription: lim_x right arrow 0 1/x^2 integral_0 ^x 2t/sqrtt^3 + 1 dt

Use l'Hospital's Rule to evaluate the limit. \(\lim _{x \rightarrow \infty} \frac{1}{x^{2}} \int_{0}^{x} \ln \left(1+e^{t}\right) d t\) Equation Transcription: Text Transcription: lim over x right arrow infinity 1/x^2 integral _0 ^x ln(1+e^t) dt

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

The Error Function The error function \(\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t\) is used in probability, statistics, and engineering. (a) Show that \(\int_{a}^{b} e^{-t^{2}} d t=\frac{1}{2} \sqrt{\pi}[\operatorname{erf}(b)-\operatorname{erf}(a)]\) (b) Show that the function \(y=e^{x^{2}} \operatorname{erf}(x)\)satisfies the differential equation \(y^{\prime}=2 x y+2 / \sqrt{\pi}\). Equation Transcription: ’ Text Transcription: erf(x)=2/sqrt pi integral _0 ^x e^-t^2 dt integral _a ^b e^-t^2 dt=1/2 sqrt pi [erf(b)-erf(a)] y=e^x^2 erf(x) y prime=2xy+2/sqrt pi

The Fresnel Function The Fresnel function S was defined in Example 3 and graphed in Figures 7 and 8. (a) At what values of x does this function have local maximum values? (b) On what intervals is the function concave upward? (c) Use a graph to solve the following equation correct to two decimal places: \(\int_{0}^{x} \sin \left(\pi t^{2} / 2\right) d t=0.2\)

The Sine Integral Function The sine integral function \(\operatorname{Si}(\mathrm{x})=\int_{0}^{x} \frac{\sin t}{t} \mathrm{dt}\) is important in electrical engineering. [The integrand f(t) = (sin t)/t is not defined when t = 0, but we know that its limit is 1 when \(t \rightarrow 0\). So we define f(0) = 1 and this makes f a continuous function everywhere.] (a) Draw the graph of Si. (b) At what values of x does this function have local maximum values? (c) Find the coordinates of the first inflection point to the right of the origin. (d) Does this function have horizontal asymptotes? (e) Solve the following equation correct to one decimal place: \(\int_{0}^{x} \frac{\sin t}{t} d t=1\)

Let \(g(x)=\int_{0}^{x} f(t) d t\), where \(f\) is the function whose graph is shown. (a) At what values of \(x\) do the local maximum and minimum values of \(g\) occur? (b) Where does \(g\) attain its absolute maximum value? (c) On what intervals is \(g\) concave downward? ________________ Equation Transcription: ________________ Text Transcription: g(x)=int_{0}^{x} f(t) d t f x g g g

Let \(g(x)=\int_{0}^{x} f(t) d t\), where \(f\) is the function whose graph is shown. (a) At what values of \(x\) do the local maximum and minimum values of \(g\) occur? (b) Where does \(g\) attain its absolute maximum value? (c) On what intervals is \(g\) concave downward? ________________ Equation Transcription: ________________ Text Transcription: g(x)=int_{0}^{x} f(t) d t f x g g g

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0,1]. \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{i^{4}}{n^{5}}+\frac{i}{n^{2}}\right)\) ________________ Equation Transcription: Text Transcription: lim _{n rightarrow infty} sum_{i=1}^{n}(frac{i^{4}}{n^{5}} + frac{i}{n^{2}})

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

Justify (3) for the case \(h<0\)

If \(f\) is continuous and \(g\) and \(h\) are differentiable functions, show that \(\frac{d}{d x} \int_{g(x)}^{h(x)} f(t) d t=f(h(x)) h^{\prime}(x)-f(g(x)) g^{\prime}(x)\) ________________ Equation Transcription: Text Transcription: f g h frac{d}{d x} int_{g(x)}^{h(x)} f(t) dt = f(h(x)) h’(x)-f(g(x)) g’(x)

(a) Show that \(1 \leqslant \sqrt{1+x^{3}} \leqslant 1+x^{3}\) for \(x \geqslant 0\). (b) Show that \(1 \leqslant \int_{0}^{1} \sqrt{1+x^{3}} d x \leqslant 1.25\). ________________ Equation Transcription: ? ? for ? 0. ? ? Text Transcription: 1 leqslant sqrt{1+x^{3}} leqslant 1+x^{3} for x geqslant 0 1 leqslant int_{0}^{1} \sqrt{1+x^{3}} d x leqslant 1.25

(a) Show that \(\cos \left(x^{2}\right) \geqslant \cos x\) for \(0 \leqslant x \leqslant 1\). (b) Deduce that \(\int_{0}^{\pi / 6} \cos \left(x^{2}\right) d x \geqslant \frac{1}{2}\). Equation Transcription: ? ? ? 1 ? Text Transcription: cos (x^2) geqslant cos x 0 leqslant x leqslant 1 integral _0 ^pi/6 cos(x^2) dx geqslant 1/2

Show that \(0 \leqslant \int_{5}^{10} \frac{x^{2}}{x^{4}+x^{2}+1} d x \leqslant 0.1\) by comparing the integrand to a simpler function.

Let \(f(x)= \begin{cases}0 & \text { if } x<0 \\ x & \text { if } 0 \leqslant x \leqslant 1 \\ 2-x & \text { if } 1<x \leqslant 2 \\ 0 & \text { if } x>2\end{cases}\) and \(g(x)=\int_{0}^{x} f(t) d t\) (a) Find an expression for g(x) similar to the one for f(x). (b) Sketch the graphs of f and g. (c) Where is f differentiable? Where is g differentiable?

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

The area labeled \(B\) is three times the area labeled \(A\). Express \(b\) in terms of \(a\). ________________ Equation Transcription: Text Transcription: B A b a

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate \(f(t)\), where \(t\) is the time measured in months since its last overhaul. Because a fixed \(\operatorname{cost} A\) is incurred each time the machine is overhauled, the company wants to determine the optimal time \(T\) (in months) between overhauls. (a) Explain why \(\int_{0}^{t} f(s) d s\) represents the loss in value of the machine over the period of time \(t\) since the last overhaul. (b) Let \(C=C(t)\) be given by \(C(t)=\frac{1}{t}\left[A+\int_{0}^{t} f(s) d s\right]\) What does \(C\) represent and why would the company want to minimize \(C\) ? (c) Show that \(C\) has a minimum value at the numbers \(t=T\). where \(C(T)=f(T)\). ________________ Equation Transcription: Text Transcription: f(t) t A T int_{0}^{t} f(s) ds t C=C(t) C(t) = frac{1}{t} [A+\int_{0}^{t} f(s) ds] C C C t =T C(T) =f (T)