1.3 The fundamental theorem of calculus  (Page 6/11)

Consider two athletes running at variable speeds $v_1\left(t\right)$ and $v_2\left(t\right).$ The runners start and finish a race at exactly the same time. Explain why the two runners must be going the same speed at some point.

Two mountain climbers start their climb at base camp, taking two different routes, one steeper than the other, and arrive at the peak at exactly the same time. Is it necessarily true that, at some point, both climbers increased in altitude at the same rate?

To get on a certain toll road a driver has to take a card that lists the mile entrance point. The card also has a timestamp. When going to pay the toll at the exit, the driver is surprised to receive a speeding ticket along with the toll. Explain how this can happen.

Set $F\left(x\right)={\displaystyle {\int }_1^x\left(1-t\right)dt}.$ Find $F^{\prime }\text{(}2)$ and the average value of $F^{\text{′}}$ over $\left[1,2\right].$

$\frac{d}{dx}{\displaystyle {\int }_1^x{e}^{\text{−}{t}^2}dt}$

$\frac{d}{dx}{\displaystyle {\int }_1^x{e}^{\text{cos}t}dt}$

$\frac{d}{dx}{\displaystyle {\int }_3^x\sqrt{9-y^2}dy}$

$\frac{d}{dx}{\displaystyle {\int }_4^x\frac{ds}{\sqrt{16-s^2}}}$

$\frac{d}{dx}{\displaystyle {\int }_x^{2x}tdt}$

$\frac{d}{dx}{\displaystyle {\int }_0^{\sqrt{x}}tdt}$

$\frac{d}{dx}{\displaystyle {\int }_0^{\text{sin}x}\sqrt{1-t^2}dt}$

$\frac{d}{dx}{\displaystyle {\int }_{\text{cos}x}^1\sqrt{1-t^2}dt}$

$\frac{d}{dx}{\displaystyle {\int }_1^{\sqrt{x}}\frac{t^2}{1+t^4}dt}$

$\frac{d}{dx}{\displaystyle {\int }_1^{x^2}\frac{\sqrt{t}}{1+t}dt}$

$\frac{d}{dx}{\displaystyle {\int }_0^{\text{ln}x}{e}^{t}dt}$

$\frac{d}{dx}{\displaystyle {\int }_1^{e^2}\text{ln}{u}^{2}du}$

The graph of $y={\displaystyle {\int }_0^{x}f(t)dt,}$ where f is a piecewise constant function, is shown here.

1. Over which intervals is f positive? Over which intervals is it negative? Over which intervals, if any, is it equal to zero?
2. What are the maximum and minimum values of f ?
3. What is the average value of f ?

The graph of $y={\displaystyle {\int }_0^{x}f(t)dt,}$ where f is a piecewise constant function, is shown here.

1. Over which intervals is f positive? Over which intervals is it negative? Over which intervals, if any, is it equal to zero?
2. What are the maximum and minimum values of f ?
3. What is the average value of f ?

The graph of $y={\displaystyle {\int }_0^x\ell (t)dt,}$ where ℓ is a piecewise linear function, is shown here.

1. Over which intervals is ℓ positive? Over which intervals is it negative? Over which, if any, is it zero?
2. Over which intervals is ℓ increasing? Over which is it decreasing? Over which, if any, is it constant?
3. What is the average value of ℓ ?