EXAMPLE 2 The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The data in the table describe the charge Q remaining on the capacitor (measured in microcoulombs) at time t (measured in seconds after the flash goes off). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t = 0.04. [Note: The slope of the tangent line represents the electric current flowing from the capacitor to the flash bulb (measured in microamperes).] SOLUTION In the figure we plot the data and use it to sketch a curve that approximates the graph of the function. Given the points P(0.04, 67.03) and R(0.00, 100.00) on the graph, we find that the slope of the secant line PR is 100.00 - mPR = 0.00 - 0.04 The table at the left shows the results of similar calculations for the slopes of other secant lines. From this table we would expect the slope of the tangent line at t = 0.04 to lie somewhere between -741.50 and -607.00. In fact, the average of the slopes of the two closest secant lines is 0.5(-741.50 - 607.00) = So, by this method, we estimate the slope of the tangent line to be (rounded to the nearest integer). Another method is to draw an approximation to the tangent line at Pand measure the sides of the triangle ABC, as in the figure. This gives an estimate of the slope of the tangent line as JAB| BCI 80.4 - 53.6 0.06 - 0.02 = -670.

Q: Find the derivative of y In x У3 t6

A:

Q: 23. GU An advertising campaign boosted sales of Crunchy Crust frozen pizza to a peak level of So…

A:

Q: Question 2: Find the slope of the tangent line of the function y = 3x² - 4x² at the point x = 12…

A: The slope of the tangent line to the curve is equal to the derivative of the given function at a…

Q: y = 7x + e5x ; (0, 1) %3D

A:

Q: 8. ( Consider the curve defined by the equation y" = 529. %3D Use logarithmic differentiation to…

A:

Q: Q1. Write down the linear (tangent line) approximation to y = e x / 2 at the point x = 0, and…

A:

Q: 14. Find a function whose tangent has slope 4x+1 for each value of x and whose graph passes through…

A:

Q: 13. Write a sine function with a horizontal shift of h=1.

A:

Q: Find the derivative of the function. y = In (8x 3- x 2) 24x-2 8x2 - x %3D 8x-2 O 8x2 -x 24x-2 8x2…

A:

Q: Find an equation for the tangent to the curve f(x) = at (-1 - 1). %3D

A: An equation for the tangent to the curve f(x)=1x3 at (-1,-1)

Q: Find the slope of the tangent line to the exponential function at the point (0, 1). y (0, 1) -1 1

A:

Q: 1 5 Determine the equation of every tangent to the curve y = =x° - 10x that has a slope of 6

A:

Q: Use logarithmic differentiation to find for the curve defined by (r+1)"

A: Given,            x + 1y=y2x

Q: (Question 3) Use a linear approximation to estimate v16.03. Your work should show how you are using…

A: Take  y=x14 as a function for approximation of the given value 16.034

Q: d) 4x-e¹ = 2y at the y-intercept

A: We are asked to find the equation of tangent line at the given value.

Q: (メ-(S/)) sp=ス Sooこイ Plot the following function

A:

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: