In Exercises 26–30, use the difference quotient definition of the derivative to find a formula for the derivative of the function.

Q: a. Simplify: eln 3 b. Use your simplification from part (a) to rewrite 3x in terms of base e.

A: Fomulae used: eln(x) = x ln(ax) = xln(a)

Q: Apply Rolle's theorem for the function f(x)=8x^2-32x+24 over the interval [1, 3].

A:

Q: Define Clairaut equation?

A: Clairaut's equation is a particular case of the Lagrange differential equation. In mathematical anal...

Q: Convert the polar equation θ =π/3 to a rectangular equation. Then use a rectangular coordinate syste...

A: Given θ=π3

Q: Rationalize the denominator : 14 / (√7 - √5)

A: Solution:-  Rationalize the denominator  A=147-5  

Q: Write an equation in point-slope form and slope-intercept form for the line passing through (-2, -6)...

A: x-3y+9=0 can be rewritten as  y=13x+3 is a line which is perpendicular to line passing from -2,-6

Q: In Exercises 53-58, compute the derivatives indicated.

A: Here given as fx, y=3x2y-6xy4 Now ∂f∂x=∂∂x3x2y-6xy4=6xy-6y4∂2f∂x2=∂∂x6xy-6y4⇒∂2f∂x2=6y

Q: Given that u = sin-1 12 /13 , evaluate cos u, tan u, cot u, sec u, and csc u.

A: given that  u=sin-11213

Q: Suppose that a population that is growing exponentially increases from 800,000 people in 2010 to 1,0...

A: To find the exponential growth function

Q: B2(a): Using the definition of derivative, find the derivative of f(x) = x² + 7.

A:

Q: Determine whether the function

A: Solution: Consider the given function is  f(x) = x8-e14x    x<014x-1       x≥0 To check whether t...

Q: find the volume of the solid generated by revolving the shaded region about the given axis.

A:

Q: Complete the following sentences in terms of a limit. a. A function is continuous from the left at a...

A: the function is defined and continuous at each and every point on that interval and a and b are fini...

Q: 1 Products Evaluate the product, and write the result in une 1orm a + bi. 31. (6 + 5i)(2 – 3i) 32. (...

A: Since you have posted a question with multiple sub-parts, we will solve only first three sub-parts f...

Q: Evaluate the integral. integral    1     /   x   (x 2 +  4) dx

A: ∫1xx2+4dx  

Q: A new auditorium is built with a foundation in the shape of one-fourth of a circle of radius 50 feet...

A: It is given that radius of circle is 50 feet.  The equation of the region is x2+y2=502; x≥0 and y≥0....

Q: Find the following using the table below. 1 2 3 3 2 4 f(x) f'(x) 9(x) (2),6 4 1 3 1 4 3 4 2 1 1 2 3 ...

A:

Q: An observer stands 300 ft from the launch site of a hot-air balloon at an elevation equal to the ele...

A: Solution:-    let observer is stand y=300 ft away from the launch site. and height of balloon from t...

Q: A new running track is to be constructed in the shape of a rectangle with semicircles at each end. T...

A: the total area of the region is covered by a rectangle and 2 semi-circles  we have been given the le...

Q: Tangent lines Carry out the following steps.a. Verify that the given point lies on the curve.b. Dete...

A: Given function is x4-x2y+y4=1 ....1 and Point=-1,1

Q: Calculus Question

A: Solution to the first question: Given,            limx→5g(x) = 10  and limx→5h(x) =28   

Q: Fill in the blank/s as given:

A: To fill the blanks in given statement 

Q: Determine whether the statement is true or false, 'Some parabolas that open to the right have equati...

A: The parabolas open to the right are symmetric about x-axis or parallel to x-axis. So vertical line c...

Q: You have a case of sweet cherries that weighs 5 kilograms. You are making Cherry Pies, and each pie ...

A: Solution:-  here  1 case of sweet cherries weighs=5 kg=5000 gm 1 pie required=1 quart  1 quart = 4 c...

Q: What is the decay rate of the radioactive substance?

A: Decay Rate of radioactive substances - Decay rate of radioactive substances is the decrease in numbe...

Q: Does the given graph (see attached) represent function that has aninverse function?

A:

Q: B4(b): A ball is dropped from certain height, its position at time t is s(t) = –6t² + 24 find the ti...

A: Given- st=-6t2+24 To find- The time in which the ball hits the ground. Concept Used- When ball hits ...

Q: Determine the convergence or divergence of the series.

A:    Definition used -            P- series convergence test-            For a series ∑n=1∞1np        ...

Q: Evaluate the integrals ∫8 cot4 t dt

A: The given indefinite integral is: ∫8 cot4t dt Since 8 is a constant, so take it outside. ∫8 cot4t dt...

Q: ): Apply Rolle's theorem for the function f(x) = 9x² – 36x + 27 over the interval [1, 3]. Justify yo...

A: Given :- Apply Rolle's Theorem for the function fx = 9x2 -36x + 27 over the integral 1,3.

Q: What are the rules for shifting, stretching, compressing, and reflecting the graph of a function?

A: Shifting: Rule for horizontally shifting the graph of fx to the left side by a units is, fx→fx+a Rul...

Q: x-values If f(x) = g(h(x)), then f'(1) =

A:

Q: dy in term of x and y if Find dx (a) y +6x=x² (b) y-cos y=x+sinx (c) x² +4xy+3y² =3 %3D (d) 3y +2y+x...

A:

Q: Differentiate implicitly to find dy/dx.                                                             ...

A: Given: secxy+tanxy+5=0

Q: I Graphing Exponential and Logarithmic Functions Sketch the graph of the function. State the domain,...

A: Since you have posted multiple questions, we will solve only first question for you. To getremaining...

Q: A ball is dropped from certain height, its position at time t is s(t)=-4t^2+16 find the time t when ...

A: The position of the ball is given by st=-4t2+16. When the ball hits the ground, st=0.

Q: 6. [3 marks] Find the equation of the tangent line to the graph of x²y² + 3xY = 208 at the point (-4...

A: Tangent line