AP Calculus Questions and FRQ help

1. A box is to be constructed from a sheet of cardboard that is 20 cm by 60 cm, by cutting out squares of length x by x from each corner and bending up the sides. What is the maximum volume this box could have?

2. Find the amplitude and period of y = –3cos(2x + 3).  Use your calculator to graph the function and state its symmetry. Find the first positive x-intercept using your calculator’s zero function.

3. Find two functions f(x) and g(x) such that f[g(x)] = x but g[f(x)] does not equal x.

4. State the vertical, horizontal asymptotes and zeros of the rational function, f(x) = ${x^2+3x+2}/{x^2+5x+4}$. Why is there no zero at x = –1?

5. Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.

6. Let f be the function defined as follows:

f of x equals the piecewise function absolute value of the quantity x minus one end quantity plus two for x is less than one, a x squared plus b x for one is less than or equal to x is less than two, and five x minus ten for x is greater than or equal to two where a and b are constants

a. If a = 2 and b = 3, is f continuous at x = 1? Justify your answer.

b. Find a relationship between a and b for which f is continuous at x = 1. Hint: A relationship between a and b just means an equation in a and b.

c. Find a relationship between a and b so that f is continuous at x = 2.

d. Use your equations from parts (ii) and (iii) to find the values of a and b so that f is continuous at both x = 1 and also at x = 2?

e. Graph the piece function using the values of a and b that you have found. You may graph by hand or use your calculator to graph and copy and paste into the document.

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7. The twice–differentiable function f is defined for all real numbers and satisfies the following conditions:

$f(0) = 3$

$f:′(0) = 5$

$f:″(0) = 7$

a. The function g is given by $g(x) = e^{ax} + f(x)$ for all real numbers, where a is a constant.  Find g ′(0) and g ″(0) in terms of a. Show the work that leads to your answers.

b. The function h is given by $h(x) = cos(kx)[f(x)] + sin(x)$ for all real numbers, where k is a constant. Find h ′(x) and write an equation for the line tangent to the graph of h at x=0.

c. For the curve given by $4x^2 + y^2 = 48 + 2xy$ show that ${dy} / {dx} = {y – 4x} / {y – x}$.

d. For the curve given by $4x^2 + y^2 = 48 + 2xy$, find the positive y-coordinate given that the x-coordinate is 2.

e. For the curve given by $4x^2 + y^2 = 48 + 2xy$, show that there is a point P with x-coordinate 2 at which the line tangent to the curve at P is horizontal.

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