# derivative

1. ### A little confused about integrals

I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function? If an integral is the area under a curve, why do we even have to find the anti derivative...
2. ### Queries regarding Inflection Points in Curve Sketching

1. Homework Statement Let A be a set of critical points of the function f(x). Let B be a set of roots of the equation f''(x)=0. Let C be a set of points where f''(x) does not exist. It follows that B∪C=D is a set of potential inflection points of f(x). Q 1: Can there exist any inflection...
3. ### Logarithmic derivative question

1. Homework Statement 1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1) 2. Homework Equations (dy/dx)(e^x) = e^x (dy/dx)ln(e^-x) = -x ? 3. The Attempt at a Solution First I believe I put ln on each set of terms (Though I...
4. ### Finding the derivative of a revenue function

1. Homework Statement The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp? 2. Homework Equations r=pq 3. The Attempt at a Solution I...
5. ### Proof showing that if F is an antiderivative of f, then f must be continuous.

1. Homework Statement Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c). 2. The attempt at a solution I attempted a...
6. ### Conceptual trouble with derivatives with respect to Arc Length

Hi, So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble. I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...
7. ### Derivative as a rate of change exercise

1. Homework Statement A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°? This is the diagram...
8. ### Help with Wronskian Equation

1. Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t) 2. Homework Equations W(y1, y2) =...
9. ### Kinematics Acceleration question

1. Homework Statement Suppose a can, after an initial kick, moves up along a smooth hill of ice. Make a statement concerning its acceleration. A) It will travel at constant velocity with zero acceleration. B) It will have a constant acceleration up the hill, but a different constant...