Differentiate y = (x2+3x)(x−2) and find the gradient at x=1. Integrate (3x2+4x) dx and find the area under…

Question

Differentiate y = (x2+3x)(x−2) and find the gradient at x=1. Integrate (3x2+4x) dx and find the area under the curve from x=0 to x=2.

ScanSolve AI Tutor · Accepted Answer

Okay, bazuunelson, let's work through these problems.

15. (a) Differentiate $y = (x^2+3x)(x-2)$

Expand the expression for $y$ .
$y = x^2(x-2) + 3x(x-2)$ $y = x^3 - 2x^2 + 3x^2 - 6x$ $y = x^3 + x^2 - 6x$
Differentiate $y$ with respect to $x$ .
Using the power rule for differentiation, $\frac{d}{dx}(x^n) = nx^{n-1}$ :
$\frac{dy}{dx} = \frac{d}{dx}(x^3) + \frac{d}{dx}(x^2) - \frac{d}{dx}(6x)$ $\frac{dy}{dx} = 3x^2 + 2x - 6$
The derivative is $\boxed{\frac{dy}{dx} = 3x^2 + 2x - 6}$ .

15. (b) Find the gradient at $x=1$
Substitute $x=1$ into the derivative found in part (a).
$\frac{dy}{dx} \Big|_{x=1} = 3(1)^2 + 2(1) - 6$ $\frac{dy}{dx} \Big|_{x=1} = 3(1) + 2 - 6$ $\frac{dy}{dx} \Big|_{x=1} = 3 + 2 - 6$ $\frac{dy}{dx} \Big|_{x=1} = 5 - 6$ $\frac{dy}{dx} \Big|_{x=1} = -1$
The gradient at $x=1$ is $\boxed{-1}$ .

16. (a) Integrate: $\int (3x^2+4x) dx$
Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ .
$\int (3x^2+4x) dx = \int 3x^2 dx + \int 4x dx$ $= 3\left(\frac{x^{2+1}}{2+1}\right) + 4\left(\frac{x^{1+1}}{1+1}\right) + C$ $= 3\left(\frac{x^3}{3}\right) + 4\left(\frac{x^2}{2}\right) + C$
Simplify the expression.
$= x^3 + 2x^2 + C$
The integral is $\boxed{x^3 + 2x^2 + C}$ .

16. (b) Find area under curve from $x=0$ to $x=2$
Evaluate the definite integral using the result from part (a).
The area is given by $\int_0^2 (3x^2+4x) dx$ .
$= \left[x^3 + 2x^2\right]_0^2$
Substitute the upper and lower limits.
$= (2^3 + 2(2^2)) - (0^3 + 2(0^2))$ $= (8 + 2(4)) - (0 + 0)$ $= (8 + 8) - 0$ $= 16$
The area under the curve is $\boxed{16}$ .

17. (a) Solve: $2^{(x+1)} = 16$
Express 16 as a power of 2.
$16 = 2^4$
Rewrite the equation with the same base.
$2^{(x+1)} = 2^4$
Equate the exponents.
$x+1 = 4$
Solve for $x$ .
$x = 4-1$ $x = 3$
The solution is $\boxed{x=3}$ .

17. (b) Express your answer in simplest form

The answer from part (a) is $x=3$ . This is already in its simplest form.
The answer is $\boxed{3}$ .

18. (a) A bag contains 4 red and 6 blue balls. Find probability of picking red
Determine the total number of balls.
Total balls = Number of red balls + Number of blue balls
Total balls = $4 + 6 = 10$
Calculate the probability of picking a red ball.
Probability (Red) = $\frac{\text{Number of red balls}}{\text{Total number of balls}}$
Probability (Red) = $\frac{4}{10}$
Simplify the probability.
Probability (Red) = $\frac{2}{5}$
The probability of picking a red ball is $\boxed{\frac{2}{5}}$ .

18. (b) If two balls are picked, find probability both are red
Calculate the probability of the first ball being red.
Probability (1st Red) = $\frac{4}{10}$
Calculate the probability of the second ball being red, given the first was red (without replacement).
After picking one red ball, there are 3 red balls left and a total of 9 balls.
Probability (2nd Red | 1st Red) = $\frac{3}{9}$
Multiply the probabilities to find the probability of both being red.
Probability (Both Red) = Probability (1st Red) $\times$ Probability (2nd Red | 1st Red)
$= \frac{4}{10} \times \frac{3}{9}$ $= \frac{2}{5} \times \frac{1}{3}$ $= \frac{2}{15}$
The probability of picking two red balls is $\boxed{\frac{2}{15}}$ .

19. (a) Find the equation of a circle with center $(1,-2)$ and radius 4
Recall the standard equation of a circle.
The equation of a circle with center $(h,k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$ .
Substitute the given values.
Given center $(h,k) = (1,-2)$ and radius $r = 4$ .
$(x-1)^2 + (y-(-2))^2 = 4^2$ $(x-1)^2 + (y+2)^2 = 16$
The equation of the circle is $\boxed{(x-1)^2 + (y+2)^2 = 16}$ .

19. (b) Find the area of the circle
Recall the formula for the area of a circle.
Area $A = \pi r^2$ .
Substitute the given radius $r=4$ .
$A = \pi (4)^2$ $A = 16\pi$
The area of the circle is $\boxed{16\pi \text{ units}^2}$ .

20. (a) A sequence has first term 5 and common difference 3. Find the 15th term
Identify the type of sequence and the given values.
This is an arithmetic sequence with first term $a_1 = 5$ and common difference $d = 3$ . We need to find the 15th term, so $n=15$ .
Use the formula for the $n$ -th term of an arithmetic sequence.
$a_n = a_1 + (n-1)d$ $a_{15} = 5 + (15-1)3$ $a_{15} = 5 + (14)3$ $a_{15} = 5 + 42$ $a_{15} = 47$
The 15th term is $\boxed{47}$ .

20. (b) Find the sum of first 15 terms
Use the formula for the sum of the first $n$ terms of an arithmetic sequence.
We can use $S_n = \frac{n}{2}(a_1 + a_n)$ . We have $n=15$ , $a_1=5$ , and $a_{15}=47$ from part (a).
$S_{15} = \frac{15}{2}(5 + 47)$ $S_{15} = \frac{15}{2}(52)$
Calculate the sum.
$S_{15} = 15 \times 26$ $S_{15} = 390$
The sum of the first 15 terms is $\boxed{390}$ .

Point	x	y
x-intercept	0	0
Minimum	1.12	−4.06
x-intercept	2	0

Differentiate y = (x2+3x)(x−2) and find the gradient at x=1. Integrate (3x2+4x) dx and find the area under the curve from x=0 to x=2.

Step-by-step solution

Expand the expression for $y$ .

Differentiate $y$ with respect to $x$ .

Substitute $x=1$ into the derivative found in part (a).

Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ .

Simplify the expression.

Evaluate the definite integral using the result from part (a).

Substitute the upper and lower limits.

Express 16 as a power of 2.

Rewrite the equation with the same base.

Equate the exponents.

Solve for $x$ .

Determine the total number of balls.

Calculate the probability of picking a red ball.

Simplify the probability.

Calculate the probability of the first ball being red.

Calculate the probability of the second ball being red, given the first was red (without replacement).

Multiply the probabilities to find the probability of both being red.

Recall the standard equation of a circle.

Substitute the given values.

Recall the formula for the area of a circle.

Substitute the given radius $r=4$ .

Identify the type of sequence and the given values.

Use the formula for the $n$ -th term of an arithmetic sequence.

Use the formula for the sum of the first $n$ terms of an arithmetic sequence.

Calculate the sum.

Still stuck on this one?

More Mathematics Questions

Differentiate y = (x2+3x)(x−2) and find the gradient at x=1. Integrate (3x2+4x) dx and find the area under the curve from x=0 to x=2.

Step-by-step solution

Expand the expression for yyy.

Differentiate yyy with respect to xxx.

Substitute x=1x=1x=1 into the derivative found in part (a).

Apply the power rule for integration, ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C.

Simplify the expression.

Evaluate the definite integral using the result from part (a).

Substitute the upper and lower limits.

Express 16 as a power of 2.

Rewrite the equation with the same base.

Equate the exponents.

Solve for xxx.

Determine the total number of balls.

Calculate the probability of picking a red ball.

Simplify the probability.

Calculate the probability of the first ball being red.

Calculate the probability of the second ball being red, given the first was red (without replacement).

Multiply the probabilities to find the probability of both being red.

Recall the standard equation of a circle.

Substitute the given values.

Recall the formula for the area of a circle.

Substitute the given radius r=4r=4r=4.

Identify the type of sequence and the given values.

Use the formula for the nnn-th term of an arithmetic sequence.

Use the formula for the sum of the first nnn terms of an arithmetic sequence.

Calculate the sum.

Still stuck on this one?

More Mathematics Questions

Expand the expression for $y$ .

Differentiate $y$ with respect to $x$ .

Substitute $x=1$ into the derivative found in part (a).

Apply the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ .

Solve for $x$ .

Substitute the given radius $r=4$ .

Use the formula for the $n$ -th term of an arithmetic sequence.

Use the formula for the sum of the first $n$ terms of an arithmetic sequence.