Q1) Find the derivative of the function. Assume that L , U , and B are constants

Question

Q1) Find the derivative of the function. Assume that L, U, and B are constants. f(x)= Lex ? Ux2 + B

f'(x)=

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Q2) With a yearly inflation rate of 8%, prices are given by P(t) = P0(1.08)t, where P0 is the price in dollars when t = 0 and t is time in years. Suppose P0 = 1. How fast (in cents/year) are prices rising when t = 10? (Round your answer to two decimal places.)

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Find the derivative of the function. Assume that L, U, and B are constants. f(x)= Lex ? Ux2 + B f'(x)= ------------------------------------------------------------------ With a yearly inflation rate of 8%, prices are given by P(t) = P0(1.08)t, where P0 is the price in dollars when t = 0 and t is time in years. Suppose P0 = 1. How fast (in cents/year) are prices rising when t = 10? (Round your answer to two decimal places.) = ¢/yr ------------------------------------------------------------------ The value of an automobile purchased in 2009 can be approximated by the function V(t) = 28(0.88)t, where t is the time, in years, from the date of purchase, and V(t) is the value, in thousands of dollars. Evaluate V(3). (Round your answer to three decimal places.) V(3) = thousands of dollars Interpret V(3). V(3) is the value of the car after 3 years.2) V(3) is the amount by which the value of the car has decreased after 3 years. 3) V(3) is the value at which the value of the car is decreasing at the end of the 3rd year. (b) Find an expression for V'(t). V'(t) = (c) Evaluate V'(3). (Round your answer to three decimal places.) V'(3) = thousands of dollars per year Interpret V'(3). V '(3) is the value of the car after 3 years.2V '(3) is the amount by which the value of the car has decreased after 3 years. 3V '(3) is the value at which the value of the car is decreasing at the end of the 3rd year. ------------------------------------------------------------------ (a) Find the slope of the graph of f(x) = 1 ? ex at the point where it crosses the x-axis. (b) Find the equation of the tangent line to the curve at the point where it crosses the x-axis. (Let x be the independent variable and y be the dependent variable.) (c) Find the equation of the line perpendicular to the tangent line at the point where it cross the x-axis. (This is thenormal line. Let x be the independent variable and y be the dependent variable.) ------------------------------------------------------------------ For what value(s) of a are y = ax and y = 1 + x tangent at x = 0? (Enter your answers as a comma-separated list.) a = ------------------------------------------------------------------ Find the exact value of c in the figure shown below, where the line l tangent to the graph of y = 2x at (0, 1) intersects the x-axis. c=

Explanation / Answer

Q1. Lex? 2Ux

Q2. P' = t*P0 * (1.08)^(t-1)

P' = 10*1*(1.08)^9 = 19.99

Q3. a. V(3) = 28* (0.88)^3 = 19.081

b. V'(t) = 28t*(0.88)^(t-1)

c. V'(3) = 28*3*(0.88)^2 = 65.0496

Q4. a. (0,0) is the point where fx crosses x axis
f'x = -e^x therefore slope at (0,0) = -1

b. (y-0) = -1 * (x-0) >> y = -x or x + y = 0

c. line perpendicular to y=-x is y=x

Q5. f'x = ax * ln a
both lines are tangent means both will have same slope therefore f'x = 1
1 = ax * ln a at x =0 >> 1 = ln a >> a = 2.7183

Q6. slope of tangent = 2^x * ln 2 = ln 2 = 0.693
equation of tangent >> (y-1) = 0.693*(x-0) >> y = 0.693x + 1
for y = 0 >> c=-1/0.693 = -1.43678

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