Q: Which of the following statements best describes the concept of continuity for a function $$f(x) at a $$point $$x = a$$?

$$f(x) is $$continuous at $$x = a if$$ $$ \lim_{x \to a} f(x) = f(a) $

Question 1

Which of the following statements best describes the concept of continuity for a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

$$f(x) is $$continuous at $$x = a if$$ $$ \lim_{x 	o a} f(x) = f(a) $

Question 2

Evaluate the following limit using L'Hospital's Rule: $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 3

A chemical reaction's rate is modeled by $$C(t) = 5e$$^{-0.2t}$$, where C$ is the concentration in mol/L and t$ is time in hours. What is the instantaneous rate of change of concentration at t = 3$ hours?

Accepted Answer

Question 4

Suppose $$f(x) = x^3$ - 3x^2$ + 4. $$Find all critical points and determine if they correspond to local maxima, minima, or neither.

Accepted Answer

Critical points at $$x = 2 ($$local min), $$x = 0 ($$local max)

Question 5

Which of the following is the correct application of the Chain Rule for differentiating $$y = \sin(x^2$)$$?

Accepted Answer

$$y' = \cos(x^2$) \cdot 2x$$

Question 6

A company wants to design a cylindrical can that holds $$500 cm$^3 of $$liquid. What is the minimum surface area required for such a can?

Accepted Answer

When $$h = 2r$$, $$S = 3\pi r^2$

Question 7

Evaluate the definite integral $$\$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 8

Which of the following is the correct formula for the arc length of the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$$

Question 9

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V = 2t^2$ + 3t. $$What is the rate at which water is entering the tank at $$t = 5$$ minutes?

Accepted Answer

$$V'(5) = 23$$ liters/min

Question 10

Which of the following integrals can be solved using the substitution $$u = 	an x$$?

Accepted Answer

$$\int \sec^2$ x dx$$

Question 11

Find the derivative of $$y = \ln(x^2$ + 1).$$

Accepted Answer

$$y' = \frac{2x}{x^2$ + 1}$$

Question 12

A medication is administered such that its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$ mg/L. How long will it take for the concentration to drop to 2$ mg/L?

Accepted Answer

$$t = \ln(5)/0.5$$

Question 13

Which of the following is the correct statement of the Fundamental Theorem of Calculus?

Accepted Answer

If $$F is an $$antiderivative of $$f on$$ $$[a, b]$$, then $$\int_a^b f(x) dx = F(b) - F(a)$$

Question 14

Evaluate the improper integral $$\$$int_1$$^\infty \frac{1}{x^2$} dx.$$

Accepted Answer

$$1$$

Question 15

Which of the following is the correct formula for the volume of a solid of revolution generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x = a to$$ $$x = b$$?

Accepted Answer

$$V = \pi \int_a^b [f(x)]^2 dx$$

Question 16

Which of the following is the correct result for the integral $$\int x e^x dx$$?

Accepted Answer

$$x e^x - e^x + C$$

Question 17

A car's velocity is given by $$v(t) = 3t^2$ + 2t (in m/s). $$How far does the car travel between $$t = 0$$ and $$t = 2$$ seconds?

Accepted Answer

$$18 m$$

Question 18

Which of the following is the correct partial fraction decomposition for $$\frac{3x + 5}{(x + 1)(x - 2)}$$?

Accepted Answer

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

Question 19

Suppose $$y = x^2$ y + \sin y. $$Find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = \frac{2x y + \cos y}{1 - x^2$}$$

MATH 101 Calculus of One Variable: Syllabus and Course Structure

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