Q: Given $$y = x^2$ e^x$$, what is the equation of the tangent line at $$x=1$$?

$$y - e = 3e(x - 1)$$,\ \text{where}\ e = $$e^1$

Question 1

Given the function $$f(x) = (x^3$ - 8x + x)^4$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 4(x^3$ - 8x + x)^3$ (3x^2$ - 8)$$

Question 2

A cylinder has a radius $$r = 3 ($$decreasing at $$1$$ unit/sec) and height $$h = 5 ($$increasing at $$2$$ units/sec). What is the rate of change of its volume $$V = \pi r^2$ h at $$this instant?

Accepted Answer

$$\frac{dV}{dt} = 12\pi$$

Question 3

Find the equation of the tangent line to $$x - y^2$ = x^2$ - 1 at $$the point $$(1, \sqrt{2}).$$

Accepted Answer

$$y - \sqrt{2} = -1(x - 1)$$

Question 4

What are the maximum and minimum values of $$f(x) = \sqrt{x^2$ - x^3$} on$$ $$[0,1]$$?

Accepted Answer

Maximum at $$x=\frac{2}{3}$$, minimum at $$x=0$$

Question 5

A 6 ft tall person walks away from a 15 ft lamppost at 2 ft/sec. When the person is 4 ft from the lamppost, how fast is the tip of their shadow moving?

Accepted Answer

$$3 ft/sec$$

Question 6

Find the equation of the tangent line to $$x + \log(xy) = x + y at$$ $$(1,1).$$

Accepted Answer

$$y - 1 = -1(x - 1)$$

Question 7

A kite is held at a constant altitude of 30 ft. The string is let out at 2 ft/sec when the horizontal distance is 40 ft. What is the rate of change of the angle of inclination $$	heta (in $$radians/sec) at that instant?

Accepted Answer

$$\frac{1}{52}$$

Question 8

Find the maximum and minimum of $$f(x) = (x^2$ - 2x + 1) + \sqrt{x} on$$ $$[0,1].$$

Accepted Answer

Maximum at $$x=1$$, minimum at $$x=0$$

Question 9

Given $$y = x^2$ e^x$$, what is the equation of the tangent line at $$x=1$$?

Accepted Answer

$$y - e = 3e(x - 1)$$,\ 	ext{where}\ e = $$e^1$

Question 10

A rectangular pen must enclose an area of 600. Three sides cost $$10/unit$$ and the fourth side costs $$30/unit. $$What is the shape of the pen that minimizes cost?

Accepted Answer

A rectangle with the expensive side as the longer side

Question 11

The height of a firework is $$s(t) = -24t^2$ + 96t. $$What is the maximum height reached by the firework?

Accepted Answer

$$192 ft$$

Question 12

For $$s(t) = -24t^2$ + 96t$$, at what time does the firework hit the ground?

Accepted Answer

$$t = 0$$ and $$t = 4$$

Question 13

Let $$g(x) = -7x^2$ + 32x. $$Find the equation of the tangent line at $$x = -1.$$

Accepted Answer

$$y + 39 = 46(x + 1)$$

Question 14

Let $$f(x) = -2x + 2 + 2^x. $$What are the zeros of $$f(x)$$?

Accepted Answer

$$x = $$ solution to $$-2x + 2 + 2^x = 0$$

Question 15

Let $$f(x) = -2x + 2 + 2^x. $$What is the $$x-$$coordinate of the critical point?

Accepted Answer

$$x = \log_2$(2)$$

Question 16

Find the derivative of $$f(x) = \sin(\sin x).$$

Accepted Answer

$$f'(x) = \cos(\sin x) \cdot \cos x$$

Question 17

Find the derivative of $$f(x) = \log(e$$^{x}$$ 	an x).$$

Accepted Answer

$$f'(x) = 1 + \sec^2$ x$$

Question 18

Find the derivative of $$f(x) = \log(\log x).$$

Accepted Answer

$$f'(x) = \frac{1}{x \log x}$$

Question 19

If $$f(x) = (	an(x^2$ + 1))(3x^2$ - 2x)^2$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = \sec^2$(x^2 + 1$$) \cdot 2x $$(3x^2$$ - $$2x)^2$$ + \$$tan(x^2 + 1$$) \cdot $$2(3x^2 - 2x)(6x - 2$$)$

Calculus Practice Assessment 2: Derivatives, Related Rates, Optimization, and Applications

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