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: A chemical reaction's concentration $$C$$ changes over time $$t$$ according to $$C(t) = 100e$$^{-0.05t}$$. What is the rate of change of concentration at t = 10$ minutes?

$$-0.05 \times 100e$$^{-0.05 \times 10}$$

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

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

Question 5

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

Accepted Answer

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

Question 6

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 7

Which of the following is the correct application of the chain rule for $$y = (3x^2$ + 1)^5$?

Accepted Answer

$$5(3x^2$$ + $$1)^4$$ \cdot 6x$$

Question 8

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

Accepted Answer

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

Question 9

A chemical reaction's concentration $$C$$ changes over time $$t$$ according to $$C(t) = 100e$$^{-0.05t}$$. What is the rate of change of concentration at t = 10$ minutes?

Accepted Answer

$$-0.05 	imes 100e$$^{-0.05 	imes 10}$$

Question 10

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

Accepted Answer

$$1$$

Question 11

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

Accepted Answer

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

Question 12

If $$y = \log_2$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x \ln(2)}$$

Question 13

A car's position is given by $$s(t) = 4t^3$ - 2t^2$ + t. $$What is the car's acceleration at $$t = 2$$ seconds?

Accepted Answer

$$20$$

Question 14

Which of the following is the correct second derivative of $$f(x) = x^4$ - 3x^2$ + 2$$?

Accepted Answer

$$12x^2 - 6$

Question 15

Find the value of $$\lim_{x 	o 0} \frac{\sin(x)}{x}.$$

Accepted Answer

$$1$$

Question 16

Which of the following functions has a point of inflection at $$x = 0$$?

Accepted Answer

$$f(x) = x^3$

Question 17

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

Accepted Answer

$$3e^{3x}$$

Question 18

A population grows according to $$P(t) = 500e$$^{0.02t}$$. What is the instantaneous rate of growth at t = 0$?

Accepted Answer

$$10$$

Question 19

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

Accepted Answer

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

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