Quiz about Differentiation

Question 1

Given $$y=5x^3+6x^2-3x+2$$ . Find $$\frac{dy}{dx}$$ .

Accepted Answer

$$15x^2+12x-3$$

Answer

$$30x^2+12$$

Answer

$$15x^2-12x+3$$

Answer

$$15x^3+12x^2-3x$$

Question 2

Given $$f\left(x\right)=\left(1+\sqrt[]{x}\right)^2$$ . Find $$\frac{dy}{dx}$$ .

Accepted Answer

$$\frac{1}{\sqrt[]{x}}+1$$

Answer

$$\frac{1}{\left(\sqrt[]{x}\right)^3+1}$$

Answer

$$x^{\frac{3}{2}}+1$$

Answer

$$2\left(1+\sqrt[]{x}\right)$$

Question 3

Find the first derivatives of $$y=\left(x^2+3\right)\left(x^2-2x\right)^4$$ .

Accepted Answer

$$\left(x^2-2x\right)^3\left(10x^3-12x^2+24x-24\right)$$

Answer

$$\left(x^2-2x\right)^3\left(4x^3-2x^2+2x-6\right)$$

Answer

$$\left(x^2-2x\right)^4\left(10x^3-12x^2+24x-24\right)$$

Answer

$$\left(x^2-2x\right)^3\left(4\left(x^2-2x\right)\left(2x-2\right)+\left(x^2-2x\right)\right)$$

Question 4

Find $$\frac{dy}{dx}$$ for the functio, $$f\left(x\right)=\frac{\operatorname{cosec}\left(x+1\right)}{e^{2x}}$$ .

Accepted Answer

$$-\frac{\operatorname{cosec}\left(x+1\right)\left[e^{2x}\cot\left(x+1\right)+2e^{2x}\right]}{e^{4x}}$$

Answer

$$\frac{1}{e^{4x}}\left(-\operatorname{cosec}\left(x+1\right)\cot\left(x+1\right)\right)-\left(\operatorname{cosec}\left(x+1\right)e^{\left(x\right)}\right)$$

Answer

$$-\frac{\operatorname{cosec}\left(x+1\right)\left[e^{2x}\cot\left(x+1\right)+2e^{2x}\right]}{e^{2x}}$$

Answer

$$e^x\left(-\operatorname{cosec}\left(x+1\right)\cot\left(x+1\right)\right)-\left(\operatorname{cosec}\left(x+1\right)e^{\left(x\right)}\right)$$

Question 5

The equation of a curve is $$f\left(x\right)=x^2-\frac{4}{x}$$ . Find the gradient of the tangent and the normal to the curve at the point where $$x=2$$ .

Accepted Answer

$$M_T=5\ \ \ M_N=-\frac{1}{5}$$

Answer

$$M_T=-\frac{1}{5}\ \ M_N=5$$

Answer

$$M_T=5\ \ M_N=-5$$

Answer

$$M_T=-5\ \ M_N=-5$$

Question 6

Find the equation of tangent of the curve $$y=\frac{12}{x^2-4x}$$ at the point where $$x=3$$ .

Accepted Answer

$$y=-\frac{8}{3}x+4$$

Answer

$$y=-\frac{8}{3}x-4$$

Answer

$$3y=8x+4$$

Answer

$$3y-8x+4=0$$

Question 7

When does f(x) decreasing?

Accepted Answer

$$-4\le x\le-\frac{4}{3}$$

Answer

$$-\infty<x\le-4$$

Answer

$$-\frac{4}{3}\le x<\infty$$

Answer

f(x) always increasing

Question 8

Given that, $$f\left(x\right)=\frac{1}{4}x^3+2x^2+4x-3$$ . What is the concavity of $$f\left(x\right)$$ on the interval $$x>0$$ ?

Accepted Answer

Concave upwards

Answer

Concave downwards

Answer

Increasing

Answer

decreasing

Question 9

$$f\left(x\right)$$ is always differentiable if it is continuous function?

Accepted Answer

False

Answer

True

Question 10

If $$\lim_{x\rightarrow a^+}\ f$$ at $$x=a$$ , we said that the function is ______.

Accepted Answer

Differentiable at $$x=a$$

Answer

Continuous at $$x=a$$

Answer

Decreasing at $$x=a$$

Answer

Concave upwards at $$x=a$$

Question 11

A curve has an equation $$y=x^3+3x^2-9x+k$$ , where $$k$$ is a constant. Find the two values of $$k$$ for which the curve has a stationary point on the x-axis.

Accepted Answer

$$k=-27,\ 5$$

Answer

$$k=-3$$

Answer

$$k=-3,\ 1$$

Answer

$$k=5$$

Question 12

Find the coordinates of the stationary points of $$f\left(x\right)=2x^3+3x^2-72x+5$$ and determine its nature.

Accepted Answer

$$\left(-4,\ 213\right)\ \max\ \left(3,\ -130\right)\ \min$$

Answer

$$\left(-4,\ 213\right)\ \min\ \left(3,\ -130\right)\ \max$$

Answer

$$\left(-4,\ 213\right)\ \min\ \left(3,\ -130\right)\ \min$$

Answer

$$\left(3,\ 213\right)\ \max\ \left(-4,\ -130\right)\ \min$$

Question 13

Given the equation of a curve, $$y=\frac{6}{\left(5-2x\right)}$$ . A point with coordinates $$\left(x,\ y\right)$$ moves along the curve in such a way that the rate of increase of $$y$$ has a constant value of $$0.02$$ unit/s. Find the rate of increase of $$x$$ when $$x=1$$ .

Accepted Answer

$$0.015$$

Answer

$$\frac{4}{3}$$

Answer

$$\frac{200}{3}$$

Answer

$$-\frac{3}{200}$$

Question 14

The diagram shows the line $$2y=x+5$$ and the curve $$y=x^2-4x+7$$ which intersect at the points $$A$$ and $$B$$ . Find the $$x-$$ coordinates of $$A$$ and $$B$$ .

Accepted Answer

$$x=3,\ 1.5$$

Answer

$$x=-3,\ 1.5$$

Answer

$$x=3,\ -1.5$$

Answer

$$x=\frac{3}{2},\ 3.5$$

Differentiation

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