Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.

t(hours)R(t)(gallons per hour)01113823510

The rate at which water leaks from a container is modeled by the twice-differentiable function R, where R(t) is measured in gallons per hour and t is measured in hours for 0≤t≤1. Values of R(t) are given in the table above for selected values of t.
(a) Use the data in the table to find an approximation for R′(12) . Show the computations that lead to your answer. Indicate units of measure.
(b)Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate ∫10R(t)ⅆt . Indicate units of measure.
(c) Use the data in the table to evaluate ∫130R′(t)ⅆt.
d) The sum ∑k=1nR(14+k2n)12n is a right Riemann sum with n subintervals of equal length. The limit of this sum as n goes to infinity can be interpreted as a definite integral. Express the limit as a definite integral.

Astudent has created the give circuit diagram. it consists of a battery, a resistor, and a light bulb. in one minute, 1.2c of charge flows through the resistor. how much charge flows through the light bulb in one minute?

The graph below shows the line of best fit for data collected on the number of cell phones in cell phone cases sold at the local electronic store on four different days

Study published in the journal of personality and individual differences found that adults with adhd displayed more creative achievement than those who didn't have the disorder. "for the same reason that adhd might create problems, like distraction, it can also allow an openness to new ideas," says holly white, assistant professor of cognitive psychology. "not being completely focused on a task lets the mind make associations that might not have happened otherwise." white and priti shah at the university of michigan gave 60 college students – half of them with adhd – a series of tests measuring creativity across 10 domains. the adhd group scored higher across the board. the adhd group showed more of a preference for brainstorming and generating ideas than the non-adhd group, which preferred refining and clarifying ideas. the adhd status of the participants was established by asking whether the individual had ever been clinically diagnosed with adhd/add. the tests of creativity were pencil-and-paper tasks administered in a laboratory setting. each of the ten scales was comprised of multiple questions, the scores on which were summed (e.g., writing creativity: "how many words can you make from the letters in the word 'psychology' invention creativity: "write down as many uses for a paper clip that you can think of.") this procedure does allow for a participant to be scored as showing no creativity under these conditions. a) state the research question in plain language (i wonder if is related to (1 point) b) state the null hypothesis (1 point) c) state the research hypothesis (1 point) d) is the research hypothesis directional or non-directional (1 point) e) name the predictor / independent variable f) give the operational definition of the predictor / independent variable g) evaluate the construct validity of the predictor / independent variable. (face, procedure, method-match) h) name the outcome / dependent variable i) give the operational definition of the outcome / dependent variable. j) evaluate the construct validity of the outcome / dependent variable. (face, procedure, method-match)