Name Date Class. Logarithmic/Exponential Differentiation and Related Rates Review AP Calculus. Find dy. dx. 1. y 4 x. y 6. 3e x.

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1 Name Date Class Find dy d. Logarithmic/Eponential Differentiation and Related Rates Review AP Calculus 1. y 4. 1 y ln. y ln 1 4. y log e y 6. y log

2 7. y e 8. e y e y e e y ln 1 e 11. y 1 e 1. y 1 e 1 y e y lne y e sin

3 d y Find d. 15. y e 16. y ln 17. e y lny y

4 19. y log e lny Let g() = f -1 (). Find g (a). f 6 a = 1. 6 f 5, 0 a = 1.

5 Refer to the table below for #-7. f() g() f () g () ½ 4 -. If h e f, find h (4). 4. If g h, find h (). ln 1 5. If h f, find h (). 6. If h g, find ' 4 h If h g f, find h ' 8. A ladder 10 ft long rests against a vertical wall and the bottom of the ladder slides away from the wall at a rate of 1 ft/sec. (a) How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? (b) At this same time, find the rate of change of the area of the right triangle formed by the ladder and the wall. (c) At this same time, at what rate at which the angle formed by the ladder and the ground decreasing? (Use radians per second)

6 9. A trough is 10 feet long and its ends have the shape of isosceles triangles that are feet across at the top and have a height of 1 feet. (a) What is the maimum amount of water the trough can hold? (b) If the trough is filled with water at the rate of 1 ft /min, how fast is the water level rising when the water is 6 inches deep? 0. Water is going into a conical cup with radius in and height of 4 in at a rate of in /min. 1 (Volume of a cone with radius r and height h is given by V r h.) (a) How fast is the height of the water in the cup increasing when the radius of the water is 1 inch? (b) How fast is the radius increasing at this time?

7 1. A spherical balloon is inflated with helium at a rate of 100 ft /min. 4 S 4r V r (a) How fast is the balloon s radius increasing at the instant the radius is 5 ft? (b) At this same instant, how fast is the surface area increasing? (c) At this same instant, what is the rate of increase of the circumference of a cross section through the center of the sphere? (d) At the moment the surface area is increasing at a rate that is numerically equivalent to the rate of the circumference, what is the radius?. A right circular cone and a hemisphere have the same base, and the cone is inscribed in the hemisphere. The figure is epanding in such a way that the combined surface area of the hemisphere and its base is increasing at a constant rate of 18 square inches per second. At what rate is the volume of the cone changing at the instant when the radius of the common base is 4 inches? Show your work. (1970 #4) Note: The surface area of a sphere of radius r is S 4r and the volume of a right circular 1 cone of height h and base radius r is V r h.

8 . Suppose that the function f has a continuous second derivative for all, and that 0 ' 0 f '' 0 0. Let g be a function whose derivative is given by f, f, g ' e f f ' for all. (1999 #4 modified) (a) Write an equation of the line normal to the graph of f at the point where = 0. (b) Given that g 0 4, write an equation of the line tangent to the graph of g at the point where = 0. 1 (c) Write the equation of the line tangent to the graph of g at the point where = 4. (d) Show that g '' e 6 f f ' f ''. Find the value of g '' 0.

AP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40

AP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40 Extreme Values in an Interval AP Calculus BC 1. The absolute maximum value of x = f ( x) x x 1 on the closed interval, 4 occurs at A) 4 B) C) 1 D) 0 E). The maximum acceleration attained on the interval