Q: Which of the following statements best describes the necessary condition for the limit $$\lim_{x \to a} f(x) to $$exist?

The left-hand and right-hand limits as $$x \to a$$ must be equal.

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 necessary condition for the limit $$\lim_{x 	o a} f(x) to $$exist?

Accepted Answer

The left-hand and right-hand limits as $$x 	o a$$ must be equal.

Question 2

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

Accepted Answer

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

Question 3

Which of the following is NOT a typical use of the derivative of a function?

Accepted Answer

Finding the area under a curve.

Question 4

Find the equation of the tangent line to $$y = 4 - x^2$ at the point (1$$, 3)$$.

Accepted Answer

$$y = -2x + 5$$

Question 5

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{7x^3$ + 2x - 1}{3x^4$ + 7x^2$ - 8x}.$$

Accepted Answer

$$0$$

Question 6

Given $$y = 6x^3$ e^x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 7

A 6 ft tall man walks away from an 18 ft lamp post at 5 ft/s. At the moment he is 45 ft from the lamp post, how fast is the length of his shadow increasing?

Accepted Answer

$$\frac{15}{6} = 2.5 ft/s$$

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$\frac{11}{(3x - 2)^2$}$$

Question 10

Suppose $$R(x) = 4x$$ and $$C(x) = 0.01x^2$ + 1.6x + 70$$ represent revenue and cost for producing $$x$$ units. What value of $$x$$ maximizes profit?

Accepted Answer

$$120$$

Question 11

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

Accepted Answer

$$-7$$

Question 12

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

Accepted Answer

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

Question 13

Find the area between $$y = x^3$ + x^2$ - 6x$$ and $$y = 6x on$$ $$[0, 3].$$

Accepted Answer

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

Question 14

Which of the following is the correct finite lower sum approximation for $$f(x) = x^2$ on $$[0, 2]$$ using 4 rectangles?

Accepted Answer

$$1.75$$

Question 15

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

Accepted Answer

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

Question 16

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 17

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

Accepted Answer

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

Question 18

Evaluate $$\int \frac{dx}{x \ln x}.$$

Accepted Answer

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

Question 19

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 20

Which of the following best describes the definite integral $$\int_a^b f(x)\ dx$$?

Accepted Answer

The net signed area between $$f(x)$$ and the $$x-$$axis from $$x = a to$$ $$x = b.$$

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