Question 1

Evaluate the following limit: $$\lim_{x 	o \frac{\pi}{2}} \frac{\cos x}{x - \frac{\pi}{2}}.$$

Accepted Answer

$$-1$$

Question 2

Find the derivative of $$y = 	an(\cos x + \sqrt{x})$$ with respect to $$x.$$

Accepted Answer

$$\sec^2$(\cos x + \sqrt{x}) \cdot (-\sin x + \frac{1}{2\sqrt{x}})$$

Question 3

Given $$x^2$$ + $$y^2 = 9$, find $$\frac{dy}{dx}$$ at the point (3$$,0)$$.

Accepted Answer

$$0$$

Question 4

Find the derivative of $$f(x) = \frac{1}{\sin^2$ x}.$$

Accepted Answer

$$-2 \cot x \csc^2$ x$$

Question 5

If $$f(x) = e$$^{x^2 + x}$$ \cos(x^2$)$$, what is $$f'(x)$$?

Accepted Answer

$$e^{x^2 + x}$$ \left[ (2x + 1) \$$cos(x^2) - 2x$$ \$$sin(x^2$$) \right]$$

Question 6

Given $$g(x) = (x^2$ + \sin x)^4$, find g$$'(x)$$.

Accepted Answer

$$4(x^2$$ + \sin $$x)^3 (2x$$ + \cos x)$$

Question 7

Find the derivative of $$y = \ln\left(\frac{x^2$}{x^2$ + 4}\right).$$

Accepted Answer

$$\frac{8}{x(x^2$ + 4)}$$

Question 8

If $$y = \sin$$^{-1}$$(\cos x)$$, for $$0 \leq y \leq \pi$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 9

A circuit is governed by $$V = IR. If$$ $$V is $$increasing at $$1$$ volt/sec and $$I is $$decreasing at $$2$$ amps/sec, when $$V = 12$$ volts and $$I = 2$$ amps, what is the rate of change of $$R$$?

Accepted Answer

$$\frac{dR}{dt} = 4$$ ohms/sec, R is increasing

Question 10

Given $$x^2$$ + $$y^2 = 100$, find $$\frac{dy}{dx}$$ and the equation of the tangent line at the point (6$$,8)$$.

Accepted Answer

$$\frac{dy}{dx} = -\frac{6}{8}$$, tangent: $$y - 8 = -\frac{6}{8}(x - 6)$$

Question 11

Approximate $$\sqrt{1001}$$ using linear approximation at $$x = 1000.$$

Accepted Answer

$$\sqrt{1001} \approx 31.016$$

Question 12

Find the derivative of $$y = \log_5$(x + 4).$$

Accepted Answer

$$\frac{1}{(x + 4) \ln 5}$$

Question 13

Find $$\frac{d}{dx}(\cos$$^{-1}$$ x)$$ for $$0 \leq x \leq \pi.$$

Accepted Answer

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

Question 14

Find the derivative of $$y = 3e$$^{x^2}$$.

Accepted Answer

$$6x e$$^{x^2}$$

Question 15

If $$4 \cos x \sin y = 1$$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{\sin x}{\cos y}$$

Question 16

Find the derivative of $$y = \cos^2$ x + \sin^2$ x.$$

Accepted Answer

$$0$$

Question 17

Find the derivative of $$y = \ln(x^2$ + 4).$$

Accepted Answer

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

Question 18

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Calculus I: Limits, Derivatives, and Applications – Study Notes

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