Q: Which of the following is the definition of the derivative of $$f(x) at$$ $$x=a$$?

$$\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}$$

Question 1

Evaluate the following limit: $$\lim\limits_{x 	o 2} (2x^2$ - 3x + 6)$$

Accepted Answer

Question 2

Evaluate the following limit: $$\lim\limits_{x 	o 0} \sin\left(\frac{5}{x-5}ight)$$

Accepted Answer

The limit does not exist

Question 3

Evaluate the following limit: $$\lim\limits_{x 	o 0} \left(\frac{\frac{1}{x} - x}{x}ight)$$

Accepted Answer

$$\infty$$

Question 4

Given the piecewise function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$, find $$\lim\limits_{x 	o 3^-} f(x).$$

Accepted Answer

$$5$$

Question 5

Evaluate the following limit: $$\lim\limits_{x 	o 1} \left(\frac{x}{x^2$-x}ight)$$

Accepted Answer

Does not exist

Question 6

Evaluate the following limit: $$\lim\limits_{x 	o 1} \left(\frac{x^2$-2}{x^3$+x}ight)$$

Accepted Answer

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

Question 7

For which values of $$x is $$the function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$ continuous? Express your answer in interval notation.

Accepted Answer

Question 8

Identify all points of discontinuity for the function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$ and determine if the discontinuity is removable.

Accepted Answer

Discontinuous at $$x=2$$ and $$x=3$$; both are not removable

Question 9

Use the Squeeze Theorem and the fact that $$-\sqrt{x} \leq \sqrt{x}\sin\left(\frac{1}{x}ight) \leq \sqrt{x}$$ for $$x\u003e0 to $$evaluate $$\lim\limits_{x 	o 0^+} \sqrt{x}\sin\left(\frac{1}{x}ight).$$

Accepted Answer

$$0$$

Question 10

Given $$f(x) is $$differentiable at $$x=2$$ and $$f(2)=9$$, $$f'(2)=7$$, what is the equation of the tangent line to $$f(x) at$$ $$x=2$$?

Accepted Answer

$$y=7(x-2)+9$$

Question 11

Suppose the position of a car is given by $$s(t) = 3t^2$ - 2t + 5. $$What is the instantaneous velocity at $$t=4$$?

Accepted Answer

$$22$$

Question 12

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of concentration at t=2$ hours?

Accepted Answer

$$-3.68$$

Question 13

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 14

Which of the following statements about limits is TRUE?

Accepted Answer

If $$f(x) is $$continuous at $$x=a$$, then $$\lim\limits_{x 	o a} f(x) = f(a).$$

Question 15

A function $$g(x) is $$defined as $$g(x) = \frac{x^2$-4}{x-2}$$ for $$x 
eq 2. $$What is $$\lim\limits_{x 	o 2} g(x)$$?

Accepted Answer

$$2$$

Question 16

Which of the following is the definition of the derivative of $$f(x) at$$ $$x=a$$?

Accepted Answer

$$\lim\limits_{h 	o 0} \frac{f(a+h)-f(a)}{h}$$

Question 17

If $$f(x) is $$continuous on $$[a,b]$$ and $$f(a)\u003c0\u003cf(b)$$, what theorem guarantees that $$f(x)=0$$ for some $$x in$$ $$(a,b)$$?

Accepted Answer

Intermediate Value Theorem

Question 18

A population of bacteria grows according to $$P(t) = 100e$$^{0.03t}$$. What is the instantaneous rate of growth at t=0$?

Accepted Answer

Question 19

Suppose $$f(x) is $$defined as $$f(x) = \frac{1}{x}$$ for $$x 
eq 0. $$Which of the following statements is true?

Accepted Answer

$$f(x) is $$discontinuous only at $$x=0.$$

Calculus I: Limits, Continuity, and Tangents – Study Notes

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