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

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

Q: Which of the following limits is used to prove $$\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$$?

$$\lim_{x \to 0} \frac{\sin x}{x}$$

Question 1

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

Accepted Answer

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

Question 2

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 3

Which rule should be used to differentiate $$y = (2x^2$ + 5x)(x^3$ - 1)$$?

Accepted Answer

Product Rule

Question 4

If $$y = \sin(x) \cdot e^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\cos(x) \cdot e^x + \sin(x) \cdot e^x$$

Question 5

Find $$\frac{d}{dx} \left( \frac{5x - 7}{2x^2$ + 3x} \right)$$ using the quotient rule.

Accepted Answer

$$\frac{(2x^2$ + 3x)(5) - (5x - 7)(4x + 3)}{(2x^2$ + 3x)^2$}$$

Question 6

Which of the following is the correct application of the chain rule for $$y = [f(x)]^n$$?

Accepted Answer

$$\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$$

Question 7

A ball is thrown upwards and its height at time $$t is $$given by $$h(t) = 20t - 5t^2$. What is the velocity of the ball at t = 2$ seconds?

Accepted Answer

$$0 m/s$$

Question 8

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 9

Find the slope of the tangent to the curve $$y = x^3$ - 2x^2$ + x at$$ $$x = 1.$$

Accepted Answer

$$1$$

Question 10

Which of the following limits is used to prove $$\lim_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\sin x}{x}$$

Question 11

Given $$y = 	an(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec^2$(x)$$

Question 12

If $$y = e$$^{3x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$3e^{3x}$$

Question 13

A chemical reaction's concentration $$C$$ changes over time $$t$$ according to $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t = 0$?

Accepted Answer

$$-1$$ units/time

Question 14

Which of the following is the correct derivative of $$y = \log_{e}(2x + 3)$$?

Accepted Answer

$$\frac{2}{2x + 3}$$

Question 15

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

Accepted Answer

$$12x^2 - 6$

Question 16

A car's position is given by $$s(t) = 3t^3$ - 2t^2$ + t. $$What is its acceleration at $$t = 2$$ seconds?

Accepted Answer

$$30$$

Question 17

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

Accepted Answer

$$\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$$

Question 18

If $$y = \sec(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec(x) 	an(x)$$

Question 19

Find $$\frac{d}{dx}(x^2$ e^x).$$

Accepted Answer

$$2x e^x + x^2$ e^x$$

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