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Chapter 3: Problem 18

The line tangent to the graph of \(f\) at \(x=3\) is \(y=4 x-2\) and the line tangent to the graph of \(g\) at \(x=3\) is \(y=-5 x+1 .\) Find the values of \((f+g)(3)\) and \((f+g)^{\prime}(3)\)

Short Answer

Expert verified
Answer: The values of (f+g)(3) and (f+g)'(3) are -4 and -1, respectively.

Step by step solution

01

Obtain f(3) and g(3) from tangent lines

We know that the tangent line of f at x = 3 is y = 4x - 2. So, we can find f(3) by plugging x = 3 into the tangent equation: \(f(3) = 4(3) - 2 = 12 - 2 = 10\) Similarly, the tangent line of g at x = 3 is y = -5x + 1. So, we can find g(3) by plugging x = 3 into the tangent equation: \(g(3) = -5(3) + 1 = -15 + 1 = -14\)
02

Obtain f'(3) and g'(3) from tangent lines

The slope of the tangent line at a point on a function's graph is equal to the derivative of the function at that point. Hence, we can find f'(3) and g'(3) from the slopes of the tangent lines: The line tangent to the graph of f at x = 3 is y = 4x - 2, which has a slope of 4. Thus, \(f'(3) = 4\). The line tangent to the graph of g at x = 3 is y = -5x + 1, which has a slope of -5. Thus, \(g'(3) = -5\).
03

Find (f+g)(3) and (f+g)'(3)

Using the values obtained in Steps 1 and 2, we can find the desired values for (f+g)(3) and (f+g)'(3): \((f+g)(3) = f(3) + g(3) = 10 + (-14) = -4\) \((f+g)'(3) = f'(3) + g'(3) = 4 + (-5) = -1\) So, the values of \((f+g)(3)\) and \((f+g)'(3)\) are -4 and -1, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative at a Point
Understanding the derivative at a point is essential in calculus, particularly when dealing with the concept of tangency. The derivative of a function at a particular value of x gives you the slope of the tangent line to the function's graph at that point.

For instance, if you have a function f, the derivative of f at x=a, often written as f'(a), tells you how steeply the graph rises or falls just when x equals a. In essence, it's the instantaneous rate of change of the function at that point. This concept is foundational in understanding how functions behave and change, contributing heavily towards the study of motion and optimization problems in calculus.
Tangent Line Slope
The concept of a tangent line slope is intrinsically linked to the derivative at a point. The slope of the tangent line is precisely the value of the derivative at the point of tangency. In practical terms, if the equation of a tangent line to a function f at a point a is given by y = mx + b, where m and b are constants, then m is equal to f'(a).

In our problem, the tangent line to f at x=3 was y=4x-2, so the slope m is 4, which means f'(3) is also 4. Similarly, for the function g with a tangent line y=-5x+1 at x=3, the slope m is -5, corresponding to g'(3).
Sum of Functions Calculus
The sum of functions calculus is a topic dealing with the addition of two or more functions to form a new function called the sum function. This is denoted as (f+g), where f and g are individual functions.

To find the sum of functions at a specific point, you simply add their values at that point. When it comes to derivatives, by the sum rule in differentiation, the derivative of the sum function at any point is equal to the sum of the derivatives of the individual functions at that point. For example, the derivative of (f+g) at x is f'(x) + g'(x). This rule allows us to combine and analyze the behavior of multiple functions simultaneously, which is vital in fields such as physics and economics, where several changing variables often interact.
Function Evaluation
The function evaluation is a simple yet fundamental process in algebra and calculus. It involves finding the output of a function given a specific input. In other words, when we have a function f(x) and we want to know what the value is at x=a, we plug a into the function in place of x and calculate the result, denoted as f(a).

For example, if we have a function f(x) = 2x^2 and we wish to find f(3), we replace x with 3 and get f(3) = 2(3)^2 = 18. In the context of our exercise, evaluating f(3) and g(3) at x=3, given their respective tangent lines, is crucial for obtaining the sum function's value at that point. It's worth noting that correct function evaluation is key to solving many problems in calculus.

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Most popular questions from this chapter

Use implicit differentiation to find\(\frac{d y}{d x}.\) $$x^{3}=\frac{x+y}{x

A mixing tank A 500 -liter (L) tank is filled with pure water. At time \(t=0,\) a salt solution begins flowing into the tank at a rate of \(5 \mathrm{L} / \mathrm{min.}\) At the same time, the (fully mixed) solution flows out of the tank at a rate of \(5.5 \mathrm{L} / \mathrm{min}\). The mass of salt in grams in the tank at any time \(t \geq 0\) is given by $$M(t)=250(1000-t)\left(1-10^{-30}(1000-t)^{10}\right)$$ and the volume of solution in the tank is given by $$V(t)=500-0.5 t$$ a. Graph the mass function and verify that \(M(0)=0\) b. Graph the volume function and verify that the tank is empty when \(t=1000 \mathrm{min}\) c. The concentration of the salt solution in the tank (in \(\mathrm{g} / \mathrm{L}\) ) is given by \(C(t)=M(t) / V(t) .\) Graph the concentration function and comment on its properties. Specifically, what are \(C(0)\) \(\underset{t \rightarrow 1000^{-}}{\operatorname{and}} C(t) ?\) d. Find the rate of change of the mass \(M^{\prime}(t),\) for \(0 \leq t \leq 1000\) e. Find the rate of change of the concentration \(C^{\prime}(t),\) for \(0 \leq t \leq 1000\) f. For what times is the concentration of the solution increasing? Decreasing?

Vibrations of a spring Suppose an object of mass \(m\) is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position \(y=0\) when the mass hangs at rest. Suppose you push the mass to a position \(y_{0}\) units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is $$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$ where \(k>0\) is a constant measuring the stiffness of the spring (the larger the value of \(k\), the stiffer the spring) and \(y\) is positive in the upward direction. Use equation (4) to answer the following questions. a. Find \(d y / d t\), the velocity of the mass. Assume \(k\) and \(m\) are constant. b. How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring? c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ( \(k\) is increased by a factor of 4 )? d. Assume y has units of meters, \(t\) has units of seconds, \(m\) has units of kg, and \(k\) has units of \(\mathrm{kg} / \mathrm{s}^{2} .\) Show that the units of the velocity in part (a) are consistent.

The total energy in megawatt-hr (MWh) used by a town is given by $$E(t)=400 t+\frac{2400}{\pi} \sin \frac{\pi t}{12}$$ where \(t \geq 0\) is measured in hours, with \(t=0\) corresponding to noon. a. Find the power, or rate of energy consumption, \(P(t)=E^{\prime}(t)\) in units of megawatts (MW). b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day? c. At what time of day is the rate of energy consumption a minimum? What is the power at that time of day? d. Sketch a graph of the power function reflecting the times when energy use is a minimum or a maximum.

Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. $$\sin y+5 x=y^{2} ;(0,0)$$ (Graph cant copy)
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