Let F(x) = Sƒ(t) dt . The graph of f on the interval [-5,4] consists of two line segments and a semi- circle, as shown below. -4 -3 -2 -1 2. 3

Q: 4. Let f(x)= a(7-a2) for all a 0 (a) Find, in terms of a, the equations of the lines tangent to…

A: Given.

Q: Let A(x) = | f(t)dt where f is the function given by the graph below. 4- 3. 2 1 -1 -1 3 4 7 9 1 2 -2…

A: Definite integration gives signed area. Using this concept, this problem can be solved easily as…

Q: 2. Let f(x) = 2e# – 3x² /a, whose graph is given to the right. Sketch in the tangent lines to f(x)…

A: PArt A: tangent line at x=1   The derivate of the tangent line at x=1, Find the derivative of the…

Q: 19. Determine whether the function R: PA(R) Ps(R) defined by R(p(x)) = (x- 2)p(x) is linear.

A:

Q: Match the graphs A, B, and C shown with the functions ƒ(x), ƒ'(x), and integral

A: Given : The graphs of f(x) , f '(x) and Anti derivative of f(x)

Q: Let F (x) S1 f (t) dt, where the graph of f is shown below: -0.5 0.5 1 1.5 --0.5. On the interval…

A: We have to find the inflection points of the function F in the interval [-0.5,1.5].

Q: Find a function f and a real number a such that :| f(t) dt = x tan-r. + 4

A:

Q: - Given that F(r) = f(t)dt, where f is defined on the interval -4, 4] as shown on the graph. On what…

A:

Q: Find a function f and a real number a such that 4 + | f(t) dt = x tan -1 x. -

A:

Q: The function f(x)= (x² – 1)³, is decreasing on the interval (- 00 , – 2) U (0,2) ^O (-1,0)U (1, ∞o )…

A:

Q: Find a function f and a real number a such that 4 : | f(t) dt = x tan- x.

A:

Q: The graph of the function below intersects at t axis on the interval [0, 4rt] at points f(t) = cos t…

A: We know that f(t) = cos(t) It cuts the t axis at the points π2,0 , 3π2 , 0 , 5π2 , 0 , ...

Q: Use the table to estimate the slope of the function at x = 1, using the secant between x = 0.8 and x…

A:

Q: Let A(x) = | f(t)dt where f is the function given by the graph below. 0. -1 -2 -3- -4+ i. A(2) = ii.…

A: Given query is to find the value of A(2) and A'(2).

Q: Find c such that the function f(2) = { S2? – 3 x c is continuous everywhere. C =

A:

Q: Let A(x) f(t)dt, with f(x) shown in the graph below. -3- ** * T?

A: As per the fundamental theorem of calculus, given a function A(x)=∫0xf(t) dt, the derivative of this…

Q: Evaluate f (t) given t = 1 for %3D 12s-13 F (s) = s(s-1)(s+2)' (11+ -) 2 e e2 o 13 37 6e2 O…

A:

Q: Find the function f(x) and all values of c such that

A: To find the function f(x) and all values of c such that∫cxf(t)dt=x2+x-2

Q: Let A(x) = | f(t)dt where f is the function given by the graph below. 2- -1 -1 3 5 7 -2 -3 A(3) =…

A:

Q: The graph of the function below intersects at t axis on the interval [0, 4Tt] at points f(t) = cos =…

A:

Q: 3. Let f(x, y) = 0, x = 0. Find fr(x, y), fy (x, y) at all points. Leave your answer as a piecewise…

A: Given: xyx2+y20,x=0,x≠0 According to the given problem: At x=0. fx(x,y)=fy(x,y)=0

Q: The harmonic function for f(z) = u(x,y) + iv(x,y) to be analytic are:

A: Given that: f=u+iv

Q: Find F as a function of x and evaluate it at x = 0, x = 1/3, and x = T/2. F(x) = cos e de F(x) =…

A:

Q: 4. Let g(x) = f6 S(1) dt, where f is the function whose graph is shown. (a) Evaluate g(0) and g(6).

A: g(x) = ∫0xf(t)dtHence g(x) is the area bounded by the curve f(t) and x-axis with sign.

Q: (b)Assume that f(x) is a function of the form -x-1; when -T < x <0 f(r) = when 0 <r < T Prove…

A: For the solution of the problem follow the next step.

Q: III. Let g be defined by the equation g(x) = dt t²-1 Show that g'(x) = secx.

A:

Q: Let A(x) | f(t)dt where f is the function given by the graph below. 2 -1 -1 -2 -3 A(1) = | A'(1) =

A:

Q: 2. Determine the relative extrema of f, if there are any, and locate any saddle points, using the…

A:

Q: Let A(r) =| f(t)dt where f is the function given by the graph below. 4+ -2 -3 А(2) - А (2) - in en

A: Given:

Q: Let A(x) = | f(t)dt where f is the function given by the graph below. 4 -1 -2 -3 -4 A(2) = A' (2) =…

A:

Q: Let f(x) = 2 – x². Over the interval [0, 2], calculate (a) R4 and (b) L4

A: As per the instruction, we will be answering only the first question. For (1), For L4:

Q: Sketch a graph of the function f(x) cos(z). -7/4 T/4 7/2 37/4 57/4 37/2 Z7/4 -1 -2-

A: We have to Sketch the given function

Q: Let A(z) = f(t)dt where f is the function given by the graph bele 3 2 1 2 3 4 5 7 1 -2 -3 -4+ A(5)=…

A:

Q: Let A(x) = | f(t)dt where f is the function given by the graph below. 5 -1 -2- -3 A(2) = A(8) = to…

A:

Q: Let the function f(x) = |x + 1| – |2 x – 1| %3D - 1.simplify the expression of f and draw…

A:

Q: The graph of the derivative f'(t) of f (t) is shown. Compute the total change of f(t) over the…

A: Solve the following

Q: Suppose that f(a) = f(b) = 0 and the second derivatives of f exist on the closed interval [a, b].…

A: Prove that ∫abx-ax-bf"xdx=2∫abfxdx.

Q: A function y(t) satisfies th

A: Given, the differential equation                       dydt=-y4-3y3+4y2

Q: Suppose h(x) = [,f(t)dt on the interval [0, 12], where f (t) is shown in the graph below. f (t) -4-…

A:

Q: 4. Are the following functions cubic splines? a. f(x) - x- 2x + 3, 0 sxs 1, - 2x - 3x2 + x + 2, 1 s…

A: Given :(a) f(x)=x3-2x+3, 0≤x≤1             =2x3-3x2+x+2, 1≤x≤2(b) f(x)=5x3-3x2+1 0≤x≤1…

Q: Let g(x)= /"rt) dt, where f is the function whose graph is shown. y 4| 3 1 4 6. 8 -1 -2 2. 2.

A: Given that       g(x) = ∫0xf(t)dt , where f(t) is a function whose graph is given. Here we have to…

Q: Let A(z) | f(t)dt where f is the function given by the graph below. -2- -3 -4+ to

A:

Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

A:

Q: 2. Consider the function f : R²R such that f(x, y) = x²y-2xy² +6xy-3y² +18y. (a) Show that f has…

A:

Q: b) Let 2z f(x) = dt t2 +t+2 find the interval at which f(x) is increasing, decreasing, concave up…

A: Given: fx=∫02x t2t2+t+2 dt To Find: To find the interval at which fx is increasing, decreasing,…

Q: Let f be a C' function on the line, and let g(x) = S f(xy)y² dy. Prove that g is a C' function and…

A: f ∈ Cn  means f is atmost n times differentiable

Q: Let A(x) = f(t)dt, with f(x) shown in the graph below.

A: Given that: Graph of f(x)

Q: Suppose h(x) = f(t)dt on the interval [0, 12], where f (t) is shown in the graph below. f (t) 8 10…

A:

Q: Use differentials and the graph of f to approximate    (a) f (1.9) and (b) f (2.04).

A:

Q: Calculate the derivatives of all orders: f'(x), f"(x), f"'(x), f(4)(x), f(x) = e3x - 1 f(x) = 3e3r-1…

A: We use chain rule to find these derivatives

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.