Q: Suppose $$f(x) = x^2$ + \sqrt{x}$$ and you want to approximate $$\sqrt{2.25}$$ using linearization at $$x = 2. $$What is the correct linear approximation?

$$\sqrt{2.25} \approx 1.5 + 0.25 \times \frac{1}{4} = 1.5625$$

Question 1

Suppose $$f(x) = x^2$ + \sqrt{x}$$ and you want to approximate $$\sqrt{2.25}$$ using linearization at $$x = 2. $$What is the correct linear approximation?

Accepted Answer

$$\sqrt{2.25} \approx 1.5 + 0.25 	imes \frac{1}{4} = 1.5625$$

Question 2

Find $$\frac{d}{dx} \left( x^2$ \sin x \right).$$

Accepted Answer

$$2x \sin x + x^2$ \cos x$$

Question 3

A biologist models the population of bacteria by $$P(t) = e$$^{2t}$$. What is the rate of change of the population at t = 1$?

Accepted Answer

$$2e^{2}$$

Question 4

Find the equation of the tangent line to $$y = \ln(x^2$ + 1) at$$ $$x = 1.$$

Accepted Answer

$$y = \ln(2) + \frac{2}{2}(x-1)$$

Question 5

If $$y = \arctan(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{1+x^2$}$$

Question 6

Find $$\frac{d}{dx} \left( \ln(\sin x) \right).$$

Accepted Answer

$$\cot x$$

Question 7

A chemical reaction rate is modeled by $$R(t) = 5e$$^{-0.2t}$$. What is R$$'(t)$$?

Accepted Answer

$$-1.0e^{-0.2t}$$

Question 8

Find $$\frac{d}{dx} \left( x^x \right).$$

Accepted Answer

$$x^x (1 + \ln x)$$

Question 9

If $$y = \sin$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{\sqrt{1-x^2$}}$$

Question 10

Find $$\frac{d}{dx} \left( 	an x ight).$$

Accepted Answer

$$\sec^2$ x$$

Question 11

A patient’s temperature is modeled by $$T(t) = 98.6 + 2\sin(\frac{\pi}{12}t)$$, where $$t is $$hours after midnight. What is the rate of change of temperature at $$t = 6$$?

Accepted Answer

$$\frac{\pi}{6} \cos(\frac{\pi}{2}) = 0$$

Question 12

Find $$\frac{d}{dx} \left( \ln(x^2$ + 3x) \right).$$

Accepted Answer

$$\frac{2x + 3}{x^2$ + 3x}$$

Question 13

If $$y = \arcsin(2x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2}{\sqrt{1-4x^2$}}$$

Question 14

Find $$\frac{d}{dx} \left( e$$^{x^2}$$ \right).$$

Accepted Answer

$$2x e$$^{x^2}$$

Question 15

Find the second derivative $$\frac{d^2$}{dx^2$} (x^3$).$$

Accepted Answer

$$6x$$

Question 16

If $$y = \ln(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2}{x}$$

Question 17

Find $$\frac{d}{dx} \left( \arccos x \right).$$

Accepted Answer

$$-\frac{1}{\sqrt{1-x^2$}}$$

Question 18

A car’s position is given by $$s(t) = t^3$ - 6t^2$ + 9t. At $$what time does the car’s velocity reach zero?

Accepted Answer

$$t = 3$$

Question 19

Find $$\frac{d}{dx} \left( \ln(x) \cdot x^2$ \right).$$

Accepted Answer

$$2x \ln x + x^2$ \cdot \frac{1}{x}$$

AP Calculus AB Unit 4: Advanced Differentiation Techniques and Applications

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