Question 1

Let $$f(x) = x^3$ - 3x^2$ + 2x$$ and $$g(x) = x^2$ - 1. $$What is the value of $$(f \circ g)(2)$$?

Accepted Answer

$$-2$$

Question 2

If $$y = x^2$ e^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$(2x + x^2$) e^x$$

Question 3

Evaluate $$\$$int_0$$^1 (3x^2$ + 2x + 1) dx.$$

Accepted Answer

$$2.5$$

Question 4

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

Accepted Answer

$$2x \cos(x^2$)$$

Question 5

A tank is being filled with water at a rate given by $$R(t) = 5t^2$ liters per minute. How much water is added between t=0$ and t=2$ minutes?

Accepted Answer

$$40$$ liters

Question 6

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

Accepted Answer

$$y = x - 1$$

Question 7

If $$\$$int_1$$^4 f(x) dx = 9$$ and $$\$$int_1$$^4 g(x) dx = 5$$, what is $$\$$int_1$$^4 [2f(x) - 3g(x)] dx$$?

Accepted Answer

$$-3$$

Question 8

A particle moves along a line so that its position at time $$t is$$ $$s(t) = t^3$ - 6t^2$ + 9t. $$What is the velocity of the particle when $$t = 2$$?

Accepted Answer

$$0$$

Question 9

Which of the following is the correct derivative of $$y = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 10

Evaluate $$\lim_{x 	o 0} \frac{\sin 3x}{x}.$$

Accepted Answer

$$3$$

Question 11

A medication is administered so that its concentration in the bloodstream at time $$t (in $$hours) is $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1}$$

Question 12

Find the area under the curve $$y = 2x$$ from $$x = 0 to$$ $$x = 3.$$

Accepted Answer

$$6$$

Question 13

If $$f(x) = x^2$ - 4x + 3$$, for which value of $$x$$ does $$f(x)$$ attain its minimum?

Accepted Answer

$$x = 2$$

Question 14

Given $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 15

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = -2x^2$ + 8x - 3$$, where $$x is $$the number of units sold in thousands. What is the maximum profit?

Accepted Answer

$$5$$ thousand dollars

Question 16

If $$\$$int_0$$^2 f(x) dx = 7$$ and $$\$$int_2$$^5 f(x) dx = 4$$, what is $$\$$int_0$$^5 f(x) dx$$?

Accepted Answer

$$11$$

Question 17

Find the value of $$\$$int_0$$^\pi \sin x dx.$$

Accepted Answer

$$0$$

Question 18

If $$y = e$$^{2x}$$, what is $$\frac{$$d^2y$$}{$$dx^2$$}$$?

Accepted Answer

$$4e^{2x}$$

Question 19

A ball is thrown upward with a velocity $$v(t) = 20 - 9.8t m/s. $$What is the maximum height reached by the ball?

Accepted Answer

$$20.4 m$$

Question 20

Evaluate $$\int x e^x dx.$$

Accepted Answer

$$x e^x - e^x + C$$

Step-by-Step Guidance for Advanced Level Calculus Exam Questions

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