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

$$2x \\sec(x^2$) \\tan^2$(x^2) + 2x$$ \\$$sec^3(x^2$)$$

$$\\frac{2 \\sin(2x)}{(1 + \\cos(2x))^2$}$$

$$\\frac{1}{\\sqrt{1 - x^2$}} - \\frac{x}{\\sqrt{1 - x^2$}} = 0$$

Question 1

Given $$y = 1 + \sqrt{x + x^2$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 2

Find $$\frac{dy}{dx}$$ for $$y = \sec(x^2$) \tan(x^2$).$$

Accepted Answer

$$2x \sec(x^2$) \tan^2$(x^2) + 2x$$ \$$sec^3(x^2$)$$

Question 3

Let $$f(x) = x^3$ + x^2$, x$$ \u003e 0$$. What is the value of $$\frac{$$d^4$$}{$$dx^4$$}f$$ at x = 2$?

Accepted Answer

$$0$$

Question 4

Find the slope of the curve $$y = y^3$ + 3 at $$the point $$(2, 1).$$

Accepted Answer

$$0$$

Question 5

Find $$dy/dx$$ for $$y = \frac{1 - \cos(2x)}{1 + \cos(2x)}.$$

Accepted Answer

$$\frac{2 \sin(2x)}{(1 + \cos(2x))^2$}$$

Question 6

At time $$t \geq 0$$, the position of a body moving along the $$x-$$axis is $$x = 3 - 2t + t^2$. At what time does the body change direction?

Accepted Answer

$$t = 1$$

Question 7

For the body in the previous question, what is its acceleration at $$t = 1$$?

Accepted Answer

$$2$$

Question 8

Find $$dy/dx$$ and simplify for $$y = \arcsin(x) + \arcsin(\sqrt{1 - x^2$})$$, $$x \u003e 0.$$

Accepted Answer

$$\frac{1}{\sqrt{1 - x^2$}} - \frac{x}{\sqrt{1 - x^2$}} = 0$$

Question 9

Find $$y(x) if$$ $$y = \arctan(2^x).$$

Accepted Answer

$$y(x) = \arctan(2^x)$$

Question 10

Find the equation of the tangent line to the curve $$x^m at$$ $$x = c.$$

Accepted Answer

$$y = mc$$^{m-1}$$(x - c) + c^m$$

Question 11

Suppose $$y(x) is $$differentiable at $$x = 3$$ and $$y(3) = 9$$, $$y'(3) = -4. $$Find $$\frac{d}{dx} \left( \sqrt{y(x)} \right) at$$ $$x = 3.$$

Accepted Answer

$$\frac{-2}{3}$$

Question 12

Find all points $$(x, y) on $$the graph of $$y = \sqrt{x - 4}$$ with tangent lines perpendicular to the line $$y = 4x + 3.$$

Accepted Answer

$$(4.25, 0.5)$$

Question 13

A chemical reaction's rate is given by $$r(t) = 5e$$^{-2t}$$. What is the instantaneous rate of change at t = 1$?

Accepted Answer

$$-10e^{-2}$$

Question 14

A medication's concentration in the bloodstream is modeled by $$C(t) = 100e$$^{-0.5t}$$. What is $$\frac{dC}{dt}$$ at t = 0$?

Accepted Answer

$$-50$$

Question 15

A ball is thrown upward and its height is $$h(t) = 20t - 5t^2$. At what time does it reach its maximum height?

Accepted Answer

$$t = 2$$

Question 16

Given $$f(x) = x^2$ e^x$$, find $$f'(x).$$

Accepted Answer

$$e^x(x^2$ + 2x)$$

Question 17

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 18

A population grows according to $$P(t) = 1000e$$^{0.03t}$$. What is the rate of growth at t = 10$?

Accepted Answer

$$30e^{0.3}$$

Question 19

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

Accepted Answer

$$\frac{2x}{1 + x^4$}$$

Calculus I: Differentiation and Applications Study Notes

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