Module 4 Lesson 1 Formative Assessment Q
1. Given $$f(x)=8-\frac{7}{x}$$ , Find all $$c$$ in the interval $$(1,7)$$ such that
$$f^{\prime}(c)=\frac{f(7)-f(1)}{7-1}$$ .
Check the Mean Value Theorem
*$$\sqrt{7}$$
$$4$$
$$\pm\sqrt{7}$$
$$\frac{7}{8}$$
2. Given the function $$f(x)=\frac{x^2+4}{x}$$ satisfies the hypothesis of the Mean Value Theorem on
the interval $$[1,4]$$ , find a number $$c$$ in the interval $$(1,4)$$ which satisfies this theorem.
Check the Mean Value Theorem
$$8$$
*$$2$$
$$3$$
$$16$$
3. Find the critical numbers of $$f(x)=x^3-12x^2$$ .
What are the properties of a critical number?
*0 and 8 only
0, 8, 12
3, 8
-8, 0, 3
4. Given that $$f(x)=-x^2+12x-28$$ has a relative maximum at $$x=6$$, choose the correct statement.
How is the derivative related to the function at extrema?
$$f^{\prime}$$ is negative on the interval $$(-\infty,6)$$
$$f^{\prime}$$ is positive on the interval $$(-\infty,\infty)$$
*$$f^{\prime}$$ is negative on the interval $$(6,\infty)$$
$$f^{\prime}$$ is positive on the interval $$(6,\infty)$$
5. Given a function defined by $$f(x)=3x^5-5x^3-8$$ , for what value(s) of $$x$$ is there a point of
relative minimum?
What determines a relative minimum?
0 and 1
0 and -1
* 1 only
1 and -1
6. Find the minimum and maximum values of $$y=3-2\sin{5x}$$
This is a periodic function, you only have to find one max and min.
* min=1; max=5
min=2; max=3
min=1; max=3
min=5; max=9
Module 4 Lesson 1 Summative Assessment Q
1. Given $$f(x)=7-\frac{6}{x}$$ , Find all $$c$$ in the interval $$(1,6)$$ such that
$$f^{\prime}(c)=\frac{f(6)-f(1)}{6-1}$$ .
*$$\sqrt{6}$$
$$\frac{6}{7}$$
$$\pm\sqrt{6}$$
$$3\sqrt{2}$$
2. Given the function $$f(x)=4+(x-1)^{\frac{1}{2}}$$ satisfies the hypothesis of the Mean Value Theorem
on the interval $$[1,5]$$ , find a number $$c$$ in the interval $$(1,5)$$ which satisfies this theorem.
1
*2
3
4
3. Find the critical numbers of $$f(x)=x^3-15x^2$$ .
-10 only
-10, 0, 5
0, 10, 15
* 0 and 10 only
4. Given that $$f(x)=-x^2+8x-15$$ has an absolute maximum at $$x=4$$ , choose the correct statement.
$$f^{\prime}$$ is negative for all real values
$$f^{\prime}$$ is positive on the interval $$(4,\infty)$$
*$$f^{\prime}$$ is negative on the interval $$(4,\infty)$$
$$f^{\prime}$$ is negative on the interval $$(-\infty,4)$$
5. Given a function defined by $$f(x)=3x^5-5x^3-8$$ , for what value(s) of $$x$$ is there a point of
relative maximum?
0 only
0 and -1
0 and 1
*-1 only
6. Find all extrema in the interval $$[0,2\pi]$$ if $$y=\sin{x}+\cos{x}$$
*$$\left(\frac{\pi}{4},\sqrt{2}\right)\text{ , }\left(\frac{5\pi}{4},-\sqrt{2}\right)$$
$$\left(\frac{\pi}{4},\sqrt{2}\right)\text{ , }\left(\frac{3\pi}{4},0\right)$$
$$\left(\frac{3\pi}{4},0\right)\text{ , }\left(\frac{5\pi}{2},-\sqrt{2}\right)$$
$$\left(0,1\right)\text{ , }\left(\pi,-1\right)$$