Q: Find the area between $$y = \sec^2$ x$$ and $$y = \cos x on$$ $$[0, \frac{\pi}{4}].$$

$$\tan \frac{\pi}{4} - \sin \frac{\pi}{4}$$

Question 1

Which of the following statements best describes the meaning of the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

It gives the instantaneous rate of change of $$f(x) at$$ $$x = a.$$

Question 2

Evaluate the following limit: $$\displaystyle \lim_{x 	o 9} \frac{\sqrt{x} - 3}{x - 9}$$

Accepted Answer

$$\frac{1}{6}$$

Question 3

A 6 ft tall man walks away from an 18 ft lamp post at a rate of 5 ft/s. How fast is the length of his shadow increasing when he is 45 ft from the lamp post?

Accepted Answer

$$\frac{15}{6} ft/s$$

Question 4

Find the derivative of $$y = 6x^3$ e^x.$$

Accepted Answer

$$18x^2 e^x$$ + $$6x^3 e^x$

Question 5

Given $$y = \frac{2x + 5}{3x - 2}$$, find $$\frac{dy}{dx}$$ using the Quotient Rule.

Accepted Answer

$$\frac{6x - 4 - 6x + 15}{(3x - 2)^2$}$$

Question 6

Find the average value of $$y = 6 - x^2$ on the interval $$[-5, 2]$$.

Accepted Answer

$$-3$$

Question 7

Evaluate $$\displaystyle \$$int_0$$^2 x^2$ dx$$ using a left Riemann sum with 4 rectangles.

Accepted Answer

$$2.25$$

Question 8

Find the exact area between $$y = x^3$ - 4x$$ and the $$x-$$axis on $$[0, 3].$$

Accepted Answer

$$\frac{81}{4}$$

Question 9

If $$\$$int_4$$^6 f(x)dx = 2$$, what is $$\$$int_6$$^4 \frac{1}{3}f(x)dx$$?

Accepted Answer

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

Question 10

Find $$\displaystyle \int \frac{dx}{x \ln x}.$$

Accepted Answer

$$\ln|\ln x| + C$$

Question 11

A box has sides $$x$$, $$y$$, $$z$$ with $$\frac{dx}{dt} = 1$$, $$\frac{dy}{dt} = 1$$, $$\frac{dz}{dt} = -2. $$Find $$\frac{dV}{dt}$$ when $$x = 4$$, $$y = 2$$, $$z = 3.$$

Accepted Answer

$$-2$$

Question 12

Find the second derivative $$\frac{d^2$y}{dt^2$}$$ for $$s = \frac{t^4$}{4} - t^3$ + t^2$.

Accepted Answer

$$3t^2 - 6t + 2$

Question 13

Evaluate $$\displaystyle \int \cos(6x - 3) dx.$$

Accepted Answer

$$\frac{1}{6}\sin(6x - 3) + C$$

Question 14

Find the area between $$y = \sec^2$ x$$ and $$y = \cos x on$$ $$[0, \frac{\pi}{4}].$$

Accepted Answer

$$	an \frac{\pi}{4} - \sin \frac{\pi}{4}$$

Question 15

Which of the following is NOT a necessary condition for the limit $$\lim_{x 	o a} f(x) to $$exist?

Accepted Answer

The function $$f(x)$$ must be defined at $$x = a.$$

Question 16

Find the derivative of $$y = e$$^{5x^3}$$ - 2x.$$

Accepted Answer

$$15x^2$$ $$e^{5x^3} - 2$

Question 17

Given $$R(x) = 4x$$ and $$C(x) = 0.01x^2$ + 1.6x + 70$$, what value of $$x$$ maximizes profit?

Accepted Answer

$$x = 120$$

Question 18

Evaluate $$\displaystyle \$$int_0$$^1 \frac{6r}{\sqrt{16 + 3r^2$}} dr.$$

Accepted Answer

$$4(\sqrt{19} - 2)$$

Question 19

Find the intervals where $$g(x) = 9x - \frac{1}{3}x^3$ is concave up.

Accepted Answer

$$(3, \infty)$$

Question 20

Find $$\frac{dy}{dx}$$ using implicit differentiation for $$2y^2 - 3xy = 8x + 4$.

Accepted Answer

$$\frac{3y + 8}{4y - 3x}$$

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