Q: Which of the following is the correct formula for the first derivative of $$f(x)$$ using the definition of the derivative?

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Question 1

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c N \u003c f(b). $$According to the Intermediate Value Theorem, which statement must be true?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f(c) = N.$$

Question 2

Use the Intermediate Value Theorem to show that the equation $$-x^2 + 4x + 1 = 0$ has a solution in the interval (-1$$, 0)$$. What is the correct reasoning?

Accepted Answer

Because $$f(-1) = -2$$ and $$f(0) = 1$$, and $$0 is $$between $$-2$$ and $$1$$, there must be $$c in$$ $$(-1, 0)$$ such that $$f(c) = 0.$$

Question 3

Which of the following is the correct formula for the first derivative of $$f(x)$$ using the definition of the derivative?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 4

Given $$f(x) = 4x^6$ - 6\sqrt{x}$$, what is $$f''(x)$$?

Accepted Answer

$$f''(x) = 120x^4$ - 3x$$^{-3/2}$$

Question 5

If $$y = f(g(x))$$, which rule is used to find $$\frac{dy}{dx}$$?

Accepted Answer

The Chain Rule

Question 6

Differentiate $$y = \sin(x^2$). $$What is $$y'$$?

Accepted Answer

$$y' = \cos(x^2$) \cdot 2x$$

Question 7

Differentiate $$y = (\sin x)^2$. What is y$$'$$?

Accepted Answer

$$y' = 2\sin x \cdot \cos x$$

Question 8

Differentiate $$y = \sqrt{x^4$ + 2x}. $$What is $$y'$$?

Accepted Answer

$$y' = \frac{1}{2}(x^4$ + 2x$$)^{-1/2}$$ \cdot (4x^3$ + 2)$$

Question 9

Given $$U(y) = \left(\frac{y^4$ + 1}{y^2$ + 1}\$$right)^5$$, what is U$$'(y)$$?

Accepted Answer

$$U'(y) = 5\left(\frac{y^4$ + 1}{y^2$ + 1}\$$right)^4$$ \cdot \frac{(y^2$ + 1)(4y^3$) - (y^4$ + 1)(2y)}{(y^2$ + 1)^2$}$$

Question 10

Differentiate $$y = (2x + 1)^5$ (x^3$ - x + 1)^4$. What is y$$'$$?

Accepted Answer

$$y' = (2x + 1)^5$ \cdot 4(x^3$ - x + 1)^3$ (3x^2$ - 1) + (x^3$ - x + 1)^4$ \cdot 5(2x + 1)^4$ \cdot 2$$

Question 11

A biologist models the population of bacteria as $$P(t) = e$$^{2t^2}$$. What is $$\frac{dP}{dt}$$?

Accepted Answer

$$\frac{dP}{dt} = e$$^{2t^2}$$ \cdot 4t$$

Question 12

A chemical reaction rate is given by $$R(x) = 	an(2x). $$What is $$R'(x)$$?

Accepted Answer

$$R'(x) = 2 \sec^2$(2x)$$

Question 13

A medical researcher models the spread of a virus as $$V(t) = \sin(e^t). $$What is $$V'(t)$$?

Accepted Answer

$$V'(t) = \cos(e^t) \cdot e^t$$

Question 14

If $$y = e$$^{\sin x}$$, what is y$$'$$?

Accepted Answer

$$y' = e$$^{\sin x}$$ \cdot \cos x$$

Question 15

A physicist models the displacement of a particle as $$s(t) = \sqrt{t^2$ + 1}. $$What is $$\frac{ds}{dt}$$?

Accepted Answer

$$\frac{ds}{dt} = \frac{t}{\sqrt{t^2$ + 1}}$$

Question 16

Which of the following is the correct application of the Chain Rule for $$y = f(g(x))$$?

Accepted Answer

$$y' = f'(g(x)) \cdot g'(x)$$

Question 17

A function $$f(x) is $$continuous on $$[1, 3]$$ with $$f(1) = -2$$ and $$f(3) = 4. $$Which value must $$f(x)$$ take for some $$x in$$ $$(1, 3)$$ according to the Intermediate Value Theorem?

Accepted Answer

$$f(x) = 0$$

Intermediate Value Theorem, Higher Derivatives, and the Chain Rule in Calculus

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