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

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: A function $$f(x) is $$said to be differentiable at $$x=a if$$:

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ exists

Q: Which of the following is a correct application of L'Hospital's Rule?

$$\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x} = \lim_{x \to 0} \frac{e^x}{1}$$

Question 1

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

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 2

Suppose $$f(x) = x^3$ - 3x^2$ + 2. $$What are the critical points of $$f(x)$$?

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 3

Which of the following statements about the Intermediate Value Theorem is correct?

Accepted Answer

If $$f(x) is $$continuous on $$[a, b]$$ and $$N is $$any number between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$[a, b]$$ such that $$f(c) = N.$$

Question 4

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 5

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)$$

Question 6

A ball is thrown upward with velocity $$v(t) = 10 - 9.8t. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 1.02$$

Question 7

Which of the following is the correct statement of the Mean Value Theorem?

Accepted Answer

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 8

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

Accepted Answer

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

Question 9

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and $$f(1) = 2$$, $$f(4) = 8. $$According to the Intermediate Value Theorem, which of the following must be true?

Accepted Answer

There exists $$c in$$ $$[1, 4]$$ such that $$f(c) = 5$$

Question 10

Evaluate $$\displaystyle \$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 11

Which of the following is the correct statement of the Fundamental Theorem of Calculus, Part I?

Accepted Answer

If $$F(x) = \int_a^x f(t) dt$$, then $$F'(x) = f(x).$$

Question 12

A function $$f(x) is $$said to be differentiable at $$x=a if$$:

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$ exists

Question 13

Which of the following is the correct formula for the chain rule?

Accepted Answer

$$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$$

Question 14

If $$f(x) = x^2$ and g(x$$) = \sin x$$, what is $$\frac{d}{dx}[f(x)g(x)]$$?

Accepted Answer

$$2x \sin x + x^2$ \cos x$$

Question 15

Which of the following is a correct application of L'Hospital's Rule?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x} = \lim_{x 	o 0} \frac{e^x}{1}$$

Question 16

Suppose $$f(x) is $$continuous and $$\$$int_1$$^3 f(x) dx = 5. $$What is $$\$$int_3$$^1 f(x) dx$$?

Accepted Answer

$$-5$$

Question 17

Which of the following is the correct formula for the derivative of $$f(x) = 	an x$$?

Accepted Answer

$$\sec^2$ x$$

Question 18

If $$f(x) = x^4$ - 4x^3$ + 6x^2$, find the intervals where f(x$$)$$ is increasing.

Accepted Answer

$$x  3$$

Question 19

Evaluate $$\displaystyle \$$int_0$$^\pi \sin x dx.$$

Accepted Answer

$$2$$

MATH 1007C Calculus I: Comprehensive Study Notes and Formula Reference

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