Question 1

Let $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 2

Which of the following statements about the function $$f(x) = \begin{cases} x^2$, & x \leq 1 \ 2x-1, & x \u003e 1 \end{cases} is $$true at $$x=1$$?

Accepted Answer

$$f is $$continuous and differentiable at $$x=1.$$

Question 3

Evaluate $$\lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 4

A function $$f is $$defined as $$f(x) = \begin{cases} 1, & x \in \mathbb{Q} \ 0, & x 
otin \mathbb{Q} \end{cases}. $$What can be said about $$\lim_{x 	o a} f(x)$$ for any real number $$a$$?

Accepted Answer

The limit does not exist.

Question 5

Suppose $$f(x) = \sqrt{x-4}. On $$which interval is $$f$$ continuous?

Accepted Answer

$$[4, \infty)$$

Question 6

Let $$f(x) = |x|. $$Which of the following is true about $$f at$$ $$x=0$$?

Accepted Answer

$$f is $$continuous but not differentiable at $$x=0.$$

Question 7

A tank in the shape of a right circular cone with height $$6 m $$and base radius $$3 m is $$being filled with water. At what rate is the water level rising when the water is $$3 m $$deep, if water is being added at $$2 m$^3/$$min?

Accepted Answer

$$\frac{1}{2\pi} m/$$min

Question 8

Evaluate $$\int \frac{1}{x^2$ + 9} dx.$$

Accepted Answer

$$\frac{1}{3} \arctan\left(\frac{x}{3}\right) + C$$

Question 9

Which of the following is the correct derivative of $$y = \ln(\sin x)$$ for $$0 \u003c x \u003c \pi$$?

Accepted Answer

$$\cot x$$

Question 10

Let $$f(x) = x^3$ - 3x^2$ + 2. $$Find all critical points and determine their nature (local max, min, or neither).

Accepted Answer

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

Question 11

A function $$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f(a) = f(b)$$, what does Rolle's Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f'(c) = 0.$$

Question 12

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

Accepted Answer

$$1$$

Question 13

Which of the following is the correct antiderivative of $$f(x) = \sec^2$ x$$?

Accepted Answer

$$	an x + C$$

Question 14

Let $$f(x) = \frac{1}{x}. $$What is the $$nth $$derivative $$f^{(n)}(x$$)$$?

Accepted Answer

$$(-1)^n n! x$$^{-n-1}$$

Question 15

A function $$f is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f(c) = 0.$$

Question 16

Evaluate $$\int x e$$^{x^2}$$ dx.$$

Accepted Answer

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

Question 17

Which of the following is the correct derivative of $$y = \arctan(x)$$?

Accepted Answer

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

Question 18

A pharmaceutical company wants to design a cylindrical can with a fixed volume $$V to $$minimize the surface area. Which equation should be solved to find the optimal ratio of height $$h to $$radius $$r$$?

Accepted Answer

$$h = 2r$$

Question 19

Let $$f(x) = x^4$ - 4x^3$. Find all inflection points.

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 20

Evaluate $$\lim_{x 	o 0^+} (\sin x)^x.$$

Accepted Answer

$$1$$

Calculus I: Functions, Limits, Continuity, and Differentiation – Structured Study Notes

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