AP Calculus BC Unit 5 Practice Test 2024-2025 PDF

MMA Math 2024-25 | AP Calculus BC Unit 5 practice test Name Block Practice Test – Unit 5 Applications of Deriv...

MMA Math 2024-25 | AP Calculus BC Unit 5 practice test Name Block Practice Test – Unit 5 Applications of Derivatives Instructions: Answer every question as best you can and show all work. Check the solutions when posted and come prepared with questions on the review day. This test also covers section 4.7 and applies unit 5 ideas to contexts explore in 4.2. 1. Determine whether each statement is true or false. I. If a function 𝑓 is continuos on a closed interval [𝑎, 𝑏], then it’s guaranteed that there is at least one value c such that 𝑓 ′ (𝑐) = 𝑓(𝑏)−𝑓(𝑎) 𝑏−𝑎. True / False II. If 𝑓(𝑎) = 5 and 𝑓(𝑏) = 5, there is at least one point in (a, b) where the graph of 𝑓 has a horizontal tangent line. True / False III. If 𝑔′ (𝑐) is a local maximum value of 𝑔′ (𝑥), then the graph of 𝑔 has an inflection point at 𝑥 = 𝑐. True / False IV. Any function that’s continuous on an open interval necessarily has a greatest and least value on the corresponding closed interval. True / False MMA Math 2024-25 | AP Calculus BC Unit 5 practice test √ 2. For the function 𝑓(𝑥) = 2 𝑥 , find all values c such that 𝑓 ′ (𝑐) equals the average rate of change on [0, 16]. 3. Calculator question. What point on the graph of 𝑦 = 12 (𝑥 − 5)2 is closest to the origin? MMA Math 2024-25 | AP Calculus BC Unit 5 practice test 4. Explain why Rolle’s theorem applies to 𝑓(𝑥) = cos 2𝑥 on the closed interval 𝜋 4 ≤ 𝑥 ≤ 3𝜋 4. 5. The position of a particle moving along a coordinate line is graphed in the figure below for 0 ≤ 𝑡 ≤ 11. The graph is composed of two line segments and a parabola, crosses the horizontal axis at 𝑡 = 32 and 𝑡 = 5.414, and has a horizontal tangent at 𝑡 = 4. (a) On what intervals is the acceleration negative? Is it ever positive? Justify your answer. (b) When is the object speeding up? Justify your answer. MMA Math 2024-25 | AP Calculus BC Unit 5 practice test 6. Find all intervals on which the graph of the given function 𝑓 is concave up and those on which the graph is concave down. 1 𝑓(𝑥) = 𝑥4 − 16𝑥2 + 8 6 7. What is the absolute maximum value of 𝑓(𝑥) = 𝑒𝑥 cos 𝑥 on [0, 1]? MMA Math 2024-25 | AP Calculus BC Unit 5 practice test 8. Show that 𝑓(𝑥) = tan 𝑥 is increasing for all values of 𝑥 for which the function is defined. 9. Evaluate the limit or show that it fails to exist. 1 + cos(2𝜋𝑥) (a) lim+ 𝑥→1 ln(2𝑥) 2 1 − 𝑒−𝑥 (b) lim 𝑥→0 sin 𝑥 MMA Math 2024-25 | AP Calculus BC Unit 5 practice test 10. The derivative of a twice-differentiable function 𝑓 is graphed in the figure below. The graph of 𝑦 = 𝑓 ′ (𝑥) has a horizontal tangent at 𝑥 = 𝑎, and is positive and increasing for 𝑥 > 𝑏. Graph of 𝑦 = 𝑓 ′ (𝑥) (a) What can be inferred about the graph of the original function, 𝑦 = 𝑓(𝑥), at 𝑥 = 𝑎? Give a reason for your answer. (b) What can be inferred about the graph of the original function, 𝑦 = 𝑓(𝑥), at 𝑥 = 𝑏? Give a reason for your answer.