AP CALCULUS AB PROBLEM SET 1
AP CALCULUS AB PROBLEM SET 1
A calculator is not allowed on these problems.
1. A particle moves along the x-axis in such a way that its position at time [pic]is given by [pic].
(a) Show that at time t = 0 the particle is moving right. Justify your answer.
(b) Find all values of t for which the particle is moving left. Justify your answer.
(c) What is the position of the particle at time t = 3?
(d) When t = 3, what is the total distance the particle has traveled? Justify your answer.
2. Let [pic].
(a) Find the x-intercepts of the graph of f. Justify your answers.
(b) Write an equation for the tangent line to the graph of f at x = 2. Show how you arrived at your answer.
(c) Write an equation of the graph that is the reflection across the y-axis of the graph of f.
3. Given[pic].
(a) Find the coordinates of the points common to the two graphs of f and g. Justify your answers.
(b) Find all zeros of f. Justify your answers.
(c) If the domain of f is limited to the closed interval [0,2], what is the range of f? Show your reasoning.
4. Let f be the real-valued function defined by [pic] .
(a) Give the domain and range of f.
(b) Determine the slope of the line tangent to the graph of f at x = 4. Justify your answers.
(c) Determine the y-intercept of the line tangent to the graph of f at x = 4. Justify your answers.
(d) Give the coordinates of the point on the graph of f where the tangent line is parallel to y = x + 12. Justify your answers.
5. Let p and q be real numbers and let f be the function defined by [pic]
(a) Find the value of q, in terms of p, for which f is continuous at x = 1. Justify your answers.
(b) Find the values of p and q for which f is differentiable at x = 1. Justify your answers.
(c) If p and q have the values determined in part (b), is f (( a continuous function? Justify your answer.
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