Q: Let $$f(x) = 6x^3$ - 3x^2$. Find all critical points of f$.

$$x = 0$$ \text{ and } x = \dfrac{1}{2}$

Question 1

Let $$T(y)$$ represent the average global temperature in year $$y$$, measured in degrees Celsius ($$^\circC). $$Given $$T(2020) = 15.01$$ and $$T'(2020) = 0.03$$ $$^\circC/$$year, what is the equation of the tangent line to $$T(y) at$$ $$y = 2020$$?

Accepted Answer

$$L(y) = 15.01 + 0.03(y - 2020)$$

Question 2

Given the function $$f(x)$$ with $$f(2) = 5$$ and the following derivative values: $$f'(0) = 1$$, $$f'(1) = 2$$, $$f'(2) = 3$$, $$f'(3) = 2$$, $$f'(4) = 1$$, what is the equation of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = 5 + 3(x - 2)$$

Question 3

Suppose the graph of $$y = g(x) is $$shown below and is twice differentiable. At $$x = 5$$, the graph has a minimum. Which of the following is true?

Accepted Answer

$$g'(5) = 0$$ and $$g''(5) \u003e 0$$

Question 4

Given the graph of $$f(x)$$ below, estimate $$f'(2).$$

Accepted Answer

$$f'(2) \approx 0$$

Question 5

Find the average rate of change of $$f$$ between $$x = 1$$ and $$x = 2 if$$ $$f(1) = 1$$ and $$f(2) = 3.$$

Accepted Answer

$$\dfrac{3 - 1}{2 - 1} = 2$$

Question 6

Suppose $$g(x) is $$differentiable at $$x = 10$$, $$g(10) = -4$$, and $$g'(10) = 0. $$Which statement must be true?

Accepted Answer

$$g(x) is $$continuous at $$x = 10$$

Question 7

Given $$f(x) = (x^2$ - 3x)\sin(x)$$, find $$f'(x)$$ using the product rule.

Accepted Answer

$$f'(x) = (2x - 3)\sin(x) + (x^2$ - 3x)\cos(x)$$

Question 8

Let $$f(x) = \dfrac{1}{x - 1}. $$Find $$f^{(4)}(x$$)$$, the fourth derivative of f$.

Accepted Answer

$$f^{(4)}(x$$) = \dfrac{24}{(x - $$1)^5$$}$$

Question 9

A car travels along a horizontal line according to $$s(t) = -t^4$ + 18t^2$ - 24t$$, where $$s is $$measured in kilometers and $$t in $$hours. What is the velocity of the car after 1 hour?

Accepted Answer

$$v(1) = s'(1) = -4(1)^3$ + 36(1) - 24 = 8 km/hr$$

Question 10

Given $$g(x) = \dfrac{2x - 2}{x + 1}$$, find $$g'(1).$$

Accepted Answer

$$g'(1) = \dfrac{4}{4} = 1$$

Question 11

Suppose $$f(x)$$ has critical points at $$x = -4, 0, 2, 5. $$The sign of $$f'(x) at $$each critical point is given in a table. At which values of $$x$$ are local maxima guaranteed to occur?

Accepted Answer

$$x = -4$$ and $$x = 2$$

Question 12

Let $$f(x) = 6x^3$ - 3x^2$. Find all critical points of f$.

Accepted Answer

$$x = 0$$ 	ext{ and } x = \dfrac{1}{2}$

Question 13

Given $$h(x) = e^x \cot(x)$$, find $$h'(x)$$ using the product rule.

Accepted Answer

$$h'(x) = e^x \cot(x) + e^x (-\csc^2$(x))$$

Question 14

Suppose $$f(x) is $$continuous and differentiable. If $$f'(x)$$ changes sign from positive to negative at $$x = a$$, what does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x)$$ has a local maximum at $$x = a$$

Question 15

Given $$f(x) = \sqrt{3x - 15\sqrt{x}}$$, find $$f'(x).$$

Accepted Answer

$$f'(x) = \dfrac{3 - \dfrac{15}{2}x$$^{-1/2}$$}{2\sqrt{3x - 15\sqrt{x}}}$$

Question 16

Suppose the graph of $$f'(x) is $$given and is positive on the interval $$(-2, 2). $$What can you conclude about $$f(x) on $$this interval?

Accepted Answer

$$f(x) is $$increasing on $$(-2, 2)$$

Question 17

Given $$f(x) = 	an(x) + \dfrac{2}{x} + x at$$ $$x = \dfrac{\pi}{2}$$, which of the following is the correct linear approximation $$L(x)$$ near $$x = \dfrac{\pi}{2}$$?

Accepted Answer

$$L(x) = f(\dfrac{\pi}{2}) + f'(\dfrac{\pi}{2})(x - \dfrac{\pi}{2})$$

Question 18

An open-top box is to be made from a $$5$$ inch by $$5$$ inch piece of plastic by removing a square from each corner and folding up the flaps. What is the maximum possible volume of the box?

Accepted Answer

$$V = 5x(5 - 2x)^2$ 	ext{ maximized for } x \approx 0.83$ inches

Calculus Study Guide: Tangent Lines, Derivatives, and Applications

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