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mathematics
calculus

Questions and Answers of Calculus

Figure 16 shows the population in millions of a developing country for the years 1900 to 1999. What is the approximate rate of change of the population in 1930? In 1900? The percentage growth is
Figure 17a and 17b show the position s as a function of time t for two particles that are moving along a line. For each particle, is the velocity increasing or decreasing? Explain.
The rate of change of electric charge with respect to time is called current. Suppose that 1/3 t3 + t coulombs of charge in amperes (coulombs per second) after 3 seconds. When will a 20-ampare fuse
The radius of a circular oil spill is growing at a constant rate of 2 kilometers per day. At what rate is the area of the spill growing 3 days after it began?
In problems 1-4 draw the tangent line to the curve through the indicated point and estimate its slope.1.2. 3. 4.
The radius of a spherical balloon is increasing at the rate of 0.25 inch per second. If the radius is 0 at time t = 0, find the rate of change in the volume at time t = 3.
Draw the graph of y = f(x) = x3 - 2x2 + 1. Then find the lope of the tangent line at (a) - 1 (b) 0 (c) 1 (d) 3.2
Draw the graph of y = f(x) = sin x sin2 2x. Then find the slope of the tangent line at (a) π/3 (b) 2.8 (c) π (d) 4.2
Consider y = x2 + 1. (a) Sketch its graph as carefully as you can. (b) Draw the tangent line at (1, 2). (c) Estimate the slope of this tangent line. (d) Calculate the slope of the secant line through
Consider y = x3 - 1. (a) Sketch its graph as carefully as you can. (b) Draw the tangent line at (2, 7) (c) Estimate the slope of this tangent line. (d) Calculate the slope of the secant line through
Find the slopes of the tangent lines to the curve y = x2 - 1 at the points where x = - 2, -1.0, 1, 2 (see Example 2).
In problem 1-4, use the definitionTo find the indicated derivative. 1. f'(1) if f(x) = x2 2. f'(2) if f(t) = 2t)2 3. f'(3) if f(t) = t2 - t 4. f'94) if f(s) = 1/s-1
In problem 1-2 use f'(x) =To find f'(x) (see Example 5). 1. F(x) = x2 - 3x 2. f(x) = x3 + 5x
In problem 1-6 the given limit is derivative, but of what function and at what point? (See Example 6).1.2. 3. 4. 5. 6.
In problem 1-4 the graph of a function y = f(x) is given. Use thi graph to sketch the graph of y = f'(x).1.2. 3. 4.
In problem 1-3, find Δy the given values of x1 and x2 (see Example 7). 1. y = 3x + 2, x1 = 1, x2 = 1.5 2. y = 3x2 + 2x+ 1, x1 = 0.0, x2 = 0.1 3. y = 1/x, x1 = 1.0, x2 = 1.2 4. y = 2/x + 1, x1 = 0,
In Problems 1-3 use f'(x) =to find the derivative at x. 1. s(x) = 2x + 1 2. f(x) = Î±x + Î² 3. r(x) 3x2 + 4
In problems 1-3, first find and simplifyThen find dy/dx by taking the limit of your answer as Î”x †’ 0. 1. y = x2 2. y = x3 - 3x2 3. y = 1 / x + 1
Sketch the graph of y = f'(x) = on - 1
Sketch the graph of y = g'(x) on -1
Consider the function y = f(x), whose graph is sketched in Figure 10.(a) Estimate f(2), f'(2), f(0.5), and f'(0.5). (b) Estimate the average rate of change in f on the interval 0.5 ‰¤ x
Figure 14 in section 2.1 shows the position s of an elevator as a function of time t. At what points does the derivative exist? Sketch the derivative of s.
Figure 15 in section 2.1 shows the normal high temperature for St. Louis. Missouri. Sketch the derivative.
Figure 11 shows two function. One is the function f, and the other is its derivative f'. Which one is which?
Figure 12 shows three functions. One is the function f; another is its derivative f', which we will call g; and the third is the derivative of g, which one is which?
Suppose that f(x + y) = f(x) f(y) for all x and y. Show that if f'(0) exists and f'(a) = f(a) f'(0).
The symmetric derivative fs(x) is defined byShow that if f'(x) exists then fs(x) exist, but that the converse is false.
Let f be differentiable and let f'(x0) = m. Find f'(- x0) if. (a) f is an odd function. (b) f is an even function.
Prove that the derivative of an odd function is an even function and the derivative of an even is an odd function.
Draw the graph of f(x) = x3 - 4x2 + 3 and its derivative f'(x) on the interval [- 2, 5] using the same axes. (a) Where on this interval is f'(x) < 0? (b) Where on this interval is f(x) decreasing as
Draw the graphs of f(x) = cos x - sin (x /2) and its derivative f'(x) on the interval [0, 9] using the same axes. (a) Where on this interval is f'(x) > 0? (b) Where on this interval is f(x)
In problem 1-6, find D, y using the rules of this section. 1. y = 2x2 2. y = 3x3 3. y = πx 4. y = πx3 5. y = 2x-2 6. y = - 3x-4
If f(0) = 4, f'(0) = -1, g(0) = - 3, and g'(0) = 5, find (a) (f ∙ g)'(0) (b) (f + g)'(0) (c) (f/g)'(0)
If f(3) = 7, f'(3) = 2, g(3) = 6, and g'(3) = - 10, find (a) (f - g) '(3) (b) (f ∙ g) '(3) (c) (g/f) '(3)
Use the product Rule to show that Dx[f(x)]2 = 2 ∙ f(x) ∙ Dx f(x).
Develop a rule for Dx[f(x) g(x) h(x)].
Find the equation of the tangent line to y = 1/(x2 + 4) at the point (1, 1/5).
Find all points on the graph of y = x3 - x2 where the tangent line is horizontal.
Find all points on the graph of y = 1/3 x3 + x2 - x where the tangent line has slope 1.
Find all points on the graph of y = 100/x5 where the tangent line is perpendicular to the line y = x.
Prove Theorem F in two ways.
The height s in feet of a ball above the ground at t seconds is given by s = - 16t2 + 40t + 100. (a) What is its instantaneous velocity at t = 2? (b) When is its instantaneous velocity 0?
There are two tangent lines to the curve y = 4x - x2 that g through (2, 5). Find the equation of both of them. Let (x0, y0) bee a point of tangency. Find two conditions that (x0, y0) must satisfy.
A space traveler is moving from left to right along the curve y = x2. When she shuts off the engine, she will continue traveling along the tangent line at the point where she is at that time. At what
A fly is crawling from left to right along the top of the curve y = 7 - x2 (Figure 5). A spider waits at the point (4, 0). Find the distance between the two insects when they first see each other.
Let P(a, b) be a point on the first quadrant portion of the curve y = 1/x and let the tangent line at P intersect the x-axis at A. Show that triangle AOP is isosceles and determine its area.
The radius of a spherical watermelon is growing at a constant rate of 2 centimeters per week. The thickness of the rind is always one-tenth of the radius. How fast is the volume of the rind growing
In problem 1-18, find Dxy. 1. y = 2 sin x + 3 cos x 2. y = sin2 x 3. y = sin2 x + cos2 x 4. y = 1 - cos2 x 5. y = sec x = 1/cos x
Find the equation of the tangent line to y = cos x at x = 1.
A Ferris wheel of radius 30 feet is rotating counterclock-wise with an angular velocity of 2 radians per second. How fast is a seat on the rim rising (in the vertical direction) when it is 15 feet
A Ferris wheel of radius 20 feet is rotating counterclock-wise with an angular velocity of 1 radian per seconds. One seat on the rim is at (20, 0) at time t = 0. (a) What are its coordinates at t =
Find the equation of the tangent line to y = tan x at x = 0.
Find all points on the graph of y = tan2 x where the tangent line is horizontal.
Find all points on the graph of y = 9 sin x cos x where the tangent line is horizontal.
Let f(x) = x - sin x. Find all points on the graph of y = f(x) where the tangent line is horizontal. Find all points on the graph y = f(x) where the tangent line has slope 2.
Show that the curves y = √2 sin x and y = √2 cos x intersect at right angles at a certain point with 0 < x < π/2.
At time t seconds, the center of a bobbing cork is 3 sin 2t centimeters above (or below) water level. What is the velocity of the cork at t = 0, π / 2, π?
Use the definition of the derivative to show that Dx(sin x2) = 2x cos x2.
Use the definition of the derivative to show that Dx(sin 5x) = 5 cos 5x.
Let f(x) = x sin x. (a) Draw the graph f(x) and f'(x) on [π, 6π]. (b) How many solutions does f(x) = 0 have on [π, 6π]? How many solutions does f'(x) = 0 have on this interval? (c) What is wrong
Let f(x) = cos3 x - 1.25 cos2 x + 0.225. Find f'(x0) at that point x0 in [π / 2, π] where f(x0) = 0.
In problem 1-5, find Dx y. 1. y = (1 + x)15 2. y = (7 + x)5 3. y = (3 - 2x)5 4. y = (4 + 2x2)7 5. y = (x3 - 2x2 + 3x + 1)11
In problem 1-3, find the indicated derivative. 1. y' where y = (x2 + 4)2 2. y' where y = (x + sin x)2 3. Dt(3t - 2/t + 5)3
In problem 1-2, evaluate the indicated derivative. 1. f' (3) if f(x) = (x2 + 1/x + 2)3 2. G'(1) if G(t) = (t2 + 9)3 (t2 - 2)4 3. F'(1) if F(t) = sin (t2 + 3t + 1)
In problems 1-3, apply the Chain Rule more than once to find the indicated derivative. 1. Dx[sin4(x2 + 3x)] 2. Dt[cos5(4t - 19)] 3. Dt[sin3 (cos t)]
In problems 1-4, use Figure 2 and 3 approximate the indicated expressions.Figure 2Figure 3 1. (f + g)' (4) 2. (f - 2g)'(2) 3. (fg)'(2) 4. (f/g)'(2)
In problem 1-4, express the indicated derivative in terms of the function F(x). Assume that F is differentiable, 1. Dx(F(2x)) 2. Dx (F(x2 + 1)) 3. Dt((F(t))-2) 4. d/dz(1/(F(z))2)
Given that F(0) = 2 and F'(0) = - 1, find G'(0) where G(x) = x / 1 + see F(2x).
Given that f(1) = 2, f'(1) = - 1, g(1) = 0 and g'(1) = 1, find F'(1) where F(x) = f(x) cos g(x).
Find the equation of the tangent line to the graph of y = 1 + x sin 3x at (π/3, 1). Where does line cross the x-axis?
Find the equation of the tangent line to y = (x2 + 1)3 (x4 + 1)2 at (1, 32).
Find the equation of the tangent line y = (x2 + 1)-2 at (1, ¼).
Where does the tangent line to y = (2x + 1)3 at (0, 1) cross the x-axis?
Where does the tangent line to y = (x2 + 1)-2 at (1, ¼) cross the x-axis?
A point P is moving in the plane so that its coordinates after t seconds are (4 cos 2t, 7 sin 2t), measured in feet. (a) Show that P is following an elliptical path. (b) Obtain an expression for L,
A wheel centered at the origin and of radius 10 centimeters is rotating counterclockwise at a rate of 4 revolutions per seconds. A point P on the rim is at (10, 0) at t = 0. (a) What are the
Consider the wheel-piston device in Figure 4. The wheel has radius 1 foot and rotates counterclockwise at 2 radians per seconds. The connecting rod is 5 feet long. The point P is at (1, 0) at time t
Do problem 70, assuming that the wheel is rotating at 60 revolutions per minute and t is measured in seconds.Consider the wheel-piston device in Figure 4. The wheel has radius 1 foot and rotates
The dial of a standard clock has a 10-centimeter radius, One end of an elastic string a attached to the rim at 12 and the other the tip of the 10-centimeter minute hand. At what rate is the string
The hour and minute hands of a clock are 6 and 8 inches long, respectively. How fast are the tips of the hands separating at 12:20 (see Figure 5).
Find the approximate time between 12:00 and 1:00 when the distance s between the tips of the hands of the clock of Figure 5 is increasing most rapidly that is, when the derivative ds/dt is largest.
Let x0 be the smallest positive value of x at which the curves y = sin x and y = sin 2x intersect. Find x0 and also the acute angle at which the two curves intersect at x0 (see problem 40 of Section
An isosceles triangle is topped by a semicircle, as shown in Figure 6. Let D be the area of triangle AOB and E be the area of the shaded region. Find a formula for D/E in terms of t and then
Show that Dx |x| = |x| / x, x ≠ 0.
In Chapter 6 we will study a function L satisfying L'(x) = 1/x. Find each of the following derivatives. (a) Dx(L(x2)) (b) Dx(L(cos4 x))
Suppose that f is a differentiable function. (a) Find d/dx f(f(x)). (b) Find d/dx f(f(x))). (c) Let f|n| denote the function defined as follows: f|1| = f and f[n] = f ͦ f[n-1] for n ≤ 2. Thus
Give a second proof of the Quotient Rule. WriteAnd use the product Rule and the Chain Rule.
In problem 1-3, find d3 y/dx3. 1. y = x3 + 3x2 + 6x 2. y = x5 + x4 3. y = (3x + 5)3
Without doing any calculating, find each derivative. (a) D4x(3x3 + 2x - 19) (b) D12x(100 x11 - 79x10) (c) D11x(x2 - 3)5
Find a formula for Dnx (1/x).
If f(x) = x3 + 3x2 - 45x - 6. Find the value of f" at each zero of f', that is, at each point c where f'(c) = 0.
Suppose that g(t) = at2 + bt + c and g(1) = 5, g'(1) = 3, and g"(1) = - 4, find a, b and c.
In problem 1-2, an object is moving along a horizontal coordinate line according to the formula s - f(t), where s, the directed distance from the origin, is in feel and t is in seconds. In each case,
If S = 1/2 t4 - 5t3 + 12t2, find the velocity of the moving object when its acceleration is zero.
If S = 1/10(t4 - 14t3 + 60t2), find the velocity of the moving object when its acceleration is zero.
Two objects move along a coordinate line. At the end of t second their directed distances from the origin, in feet, are given by s1 = 4t - 3t2 and s2 = t2 - 2t, respectively. (a) When do they have
The positions of two objects, P1 and P2, on a coordinate line at the end of t seconds are given by s1 = 3t2 - 12t2 + 18t + 5 and s2 = - t3 + 9i2 - 12t, respectively. When do the two objects have the
An object throw directly upward is at a height of s = - 16t2 + 48t + 256 feel after t seconds (see Example 4). (a) What is its initial velocity? (b) When does it reach its maximum height? (c) What is
An object thrown directly upward from ground level with an initial velocity of 48 feet per second is s = 48t - 16t2 feet high at the end of t seconds. (a) What is the maximum height attained? (b) How
A projectile is fires directly upward from the ground with an initial velocity of v0 feet per second. Its height in t seconds is given by s = v0t - 16t2 feet. What must its initial velocity be for

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