Identify the exponential function for the given graph.

t^2+4t+4 t^2-2t+1 Consider the parametric curve a. Find points on the parametric curve where the tangent lines are vertical. b.

kondaur [170]

Answer:

a) $t = 1$ , b) $t = -2$ , c) Concave upward: $(-\infty, 1)$ . No intervals being concave downward.

Step-by-step explanation:

a) Tangent line is vertical if slope is undefined, whose points are associated with discontinuities of the function. Rational functions have discontinuities associated with the denominator:

$f(t) = \frac{t^{2}+4\cdot t +4}{t^{2}-2\cdot t + 1}$

By factorizing each component, the function is re-arranged as:

$f(t) = \frac{(t+2)^{2}}{(t-1)^{2}}$

There is no avoidable discontinuities. The only point where tangent line is vertical is $t = 1$ .

b) Tangent line is horizontal if slope is equal to zero, whose point is associated to the points that makes function equal to zero. Rational functions have horizontal tangent lines if numerator is equal to zero and denominator is different to zero. Hence, the only point where tangent line is horizontal is $t = -2$ .

c) An interval is concave upward if exist an absolute minimum inside, which can be found by the help of the First and Second Derivative Tests.

$f'(t) = \frac{2\cdot (t+2)\cdot (t-1)^{2}-2\cdot (t-1)\cdot (t+2)^{2}}{(t-1)^{4}}$

$f'(t) = 2\cdot (t+2)\cdot (t-1)\cdot \left[\frac{t-1-t-2}{(t-1)^{4}}\right]$

$f'(t) = -\frac{6\cdot (t+2)\cdot (t-1)}{(t-1)^{4}}$

$f'(t) = -\frac{6\cdot (t+2)}{(t-1)^{3}}$

The only critical point is:

$-6\cdot (t+2) = 0$

$t = -2$ (which coincides with the result of point b)

$f''(t) = -6\cdot \left[\frac{(t-1)^{3}-3\cdot (t-1)^{2}\cdot (t+2)}{(t-1)^{6}}\right]$

$f''(t) = -6\cdot \left[\frac{1}{(t-1)^{3}}-\frac{3\cdot (t+2)}{(t-1)^{4}} \right]$

The value associated with the critical point is:

$f''(-2) = \frac{2}{9}$

Which means that critical point is an absolute minimum, and, consequently, the interval that is concave upward is $(-\infty, 1)$ . There is no absolute maximums and, therefore, there is no interval that is concave downward.

7 0

4 years ago

George has sold $75 worth of sandwiches at his concession stand today. Each sandwich sells for $5. He hopes to sell at least $60

boyakko [2]

5x + 75 > = 600 (thats greater then or equal)

3 0

3 years ago

Please help for 11 will give brainiest

nikklg [1K]

Answer:

5=5=10

Step-by-step explanation:

4 0

3 years ago

Kathleen, Penny, and Megan all order 12-ounce smoothies. After 5 minutes, Kathleen still has 3/4 left, Penny has 5/6 left, and M

____ [38]

Penny has the most left of her smoothie
you have to find the common denominator(number that all of the denominators are divisible into.
the common denominator is 24
convert all the fraction accordingly
Kathleen has 9/24 smoothie
penny has 20/24 smoothie
megan has 15/24 smoothie<span />

4 0

3 years ago

The statement tan theta -12/5, csc theta -13/5, and the terminal point determined by theta is in quadrant 2."

Harlamova29_29 [7]

Answer:

Answer C:

Cannot be true because $csc(\theta)$ is greater than zero in quadrant 2.

Step-by-step explanation:

When the csc of an angle is negative, since the cosecant function is defined as:

$csc(\theta)=\frac{1}{sin(\theta)}$

that means that the sin of the angle must be negative, and such cannot happen in the second quadrant. The sine function is positive in the first and second quadrant.

Therefore, the correct answer is:

Cannot be true because $csc(\theta)$ is greater than zero in quadrant 2.

6 0

4 years ago